This site is supported by donations to The OEIS Foundation.

Binary Quadratic Forms and OEIS

From OeisWiki
Jump to: navigation, search

1. Summary

  • A binary quadratic form ax^2+bxy+cy^2 has discriminant d = b^2-4ac, and is positive definite if d < 0, or indefinite if d > 0.
  • This web page gives an index to the following sequences in the OEIS:
    • numbers (or primes) represented by positive definite binary quadratic forms (Section 3),
    • numbers (or primes) represented by indefinite binary quadratic forms (Section 4),
  • This page also lists programs for computing these sequences (in Section 5) and a list of references and links (in Section 6).
  • There is a lot still to be done:
    • there are several cases where a sequence appears to arise in several different ways, not all of which have been proved to give the same sequence (see for example the multiple meanings listed in A033212 and A141184). Once these have been resolved, the entries can probably be merged. [Added Oct 18 2014: Thanks to the work of J. B. Tunnel (see the link in A033212, some of these conjectured duplicates are now known to be genuine, and have been merged. - N. J. A. Sloane]
    • not all the sequences presently in the OEIS have been included, and
    • there are many more that could be added (both to the OEIS and here).
    • Many of the sequences arising from indefinite quadratic forms (see Section 4) were computed by "brute force", which is a notoriously unreliable method (as one knows from studying Pell's equation). These also need to be checked.

2. Introduction and notation

  • The starting point is Fermat's result that a prime p is the sum of two squares if and only if p is either 2 or congruent to 1 mod 4 (see A002313), and more generally that a number n is the sum of two squares if and only if every prime congruent to 3 mod 4 occurs to an even power in n (A001481) — that is, if and only if n = square * 2^{0 or 1} * {product of distinct primes == 1 (mod 4)}.
  • An equivalent statement is that we are looking for integers x and y such that x^2+y^2 = p, or x^2+y^2 = n.
  • It is natural to ask, what happens if instead of x^2+y^2, we consider an arbitrary quadratic form, ax^2+bxy+cy^2, where a, b, and c are integers? What primes does it represent, and what numbers does it represent? These questions are easier to answer if the form is positive definite.
  • A binary quadratic form f(x,y) = ax^2+bxy+cy^2 has discriminant d = b^2-4ac, and is positive definite if d < 0, and indefinite if d > 0. Note that d is always congruent to 0 or 1 (mod 4).
    • f(x,y) represents an integer n if there are integers x and y such that f(x,y)=n [Buell, p. 1; Cox, 1st ed., p. 24; 2nd ed., p. 22].
    • Of course any quadratic form represents 0; f(x,y) represents 0 nontrivially if f(x,y)=0 has a solution with x and y not both 0.
    • f(x,y) primitively represents n if f(x,y)=n has a solution with x and y relatively prime.
  • The Gram matrix corresponding to the form is the 2 X 2 matrix [a, b/2; b/2, c], which if b is odd we may replace by [2a, b; b, 2c]. The determinant of the form is the determinant of the Gram matrix, that is, either ac-b^2/4 or 4ac-b^2.
  • One usually assumes that the discriminant is not a perfect square. For if b^2-4ac = s^2, then after multiplying by 4a and completing the square, the form decomposes into the product of two linear factors, (2ax+(b+s)y)(2ax+(b-s)y), and solving f(x,y)=n reduces to factorizing n.

3. Index to positive definite binary quadratic forms

Remarks

  • The first number in each line is the discriminant d.
  • Buell (1989, pp. 19 and 20) gives a table of these forms (with |d| <= 163 for d == 1 mod 4, |d| <= 100 for d == 0 mod 4), and there is a list classified by determinant (ac-b^2/4) rather than discriminant, for determinant <= 50, in Conway-Sloane, pages 362-363. It would be nice to have a reference to a bigger table. The MAGMA commands Q := BinaryQuadraticForms(d); ReducedForms(Q); produce a list of the reduced forms of discriminant d.
  • In the list, a pair [A111111, A222222] indicates that A111111 gives all the integers represented by the form, and A222222 just gives the primes. Pairs are separated by semicolons.
  •  ??? indicates an A-number to be filled in, probably by creating a new sequence.
  • In a number of cases there are two or more candidates for A222222 (with different definitions). It would be nice to have proofs that these sequences are in fact identical, in which case the entries should be merged. The paper by Voight (see Section 6) should completely solve this problem.
  • Primes of the form x^2+Ny^2, where N is a convenient number (A000926) are especially interesting and are listed in Section 3.4
  • In the classical theory, one allows x and y to take any integral values. But sometimes, in the list below, there are parenthetical entries for subsequences where x and y take only nonnegative values (e.g. see d=-7).
  • Obviously many more discriminants could be added to this list - please help! (I have checked that this list of quadratic forms is correct list through about d=-47.) Also many of these entries need b-files.
  • Although it is not at present a major focus of this web page, for each positive definite quadratic form, we may also consider the sequence giving the number of integer solutions (x, y) to x^2 + x*y + 2*y^2 = n (or 2n, if the form only represents even numbers). See, for example, A004016 listed at d=-3 and A002652 listed at d=-7. It would be nice to include more such sequences in this list.

The union list of all positive definite quadratic forms ax^2+bxy+cy^2

Primes of the form x^2+Ny^2, where N <= 102

  • This is a sublist of the list in Section 3.2.
  • The list:

Complete list of primes of the form x^2+Ny^2, where N is a convenient number (A000926)

  • For each N in A000926, there is a list R of numbers such that a prime p is represented by x^2+Ny^2 if and only if p is congruent mod 4N to an element of R. The lists R are given in the rows of the irregular triangle A139642. For example, if N = 4, R is {1, 5, 9, 13}, and so p is of the form x^2+4y^2 if and only if it is congruent to one of 1, 5, 9, or 13 mod 16, or, more simply, if and only if p is of the form 4k+1, which gives A002144.
  • This is a sublist of the list in Section 3.2.
  • The list:

4. Index to indefinite binary quadratic forms

Remarks

  • The first number in each line is the discriminant d.
  • There is a list of these indefinite forms, classified by determinant (ac-b^2/4) rather than discriminant, for |determinant| <= 100, in Conway-Sloane, pages 362-363. It would be nice to have a reference to a bigger table arranged by discriminant. Perhaps Ince (1934), which I do not presently have access to. The MAGMA commands Q := BinaryQuadraticForms(d); ReducedForms(Q); produce a list of the reduced forms of discriminant d.
  • Discriminants which are squares > 9 have not yet been included (but certainly should be).
  • In the list, a pair [A111111, A222222] indicates that A111111 gives all the nonnegative integers represented by the form, and A222222 just gives the primes. Pairs are separated by semicolons.
  • In a number of cases there are two candidates for A222222 (with different definitions). It would be nice to have proofs that these sequences are in fact identical, in which case the entries should be merged.
  • Note that if ax^2+bxy+cy^2 is an indefinite form, then -ax^2+bxy-cy^2, even though it has the same discriminant, in general will represent a different set of positive numbers and a different set of primes.
  • The indefinite case is more difficult than the positive definite case (see references in Section 6). For one thing, it cannot be solved by brute force.
    • Consider for example the indefinite form x^2-ky^2, for a positive integer k, with discriminant 4k. This is a special case of Pell's equation. A famous example is x^2-61y^2, with discriminant 244. Does this represent 1? Trivially x = 1, y = 0 is a solution, but the smallest solution with y positive is x = 1766319049, y = 226153980, not something that could be found by brute force. See the references in Section 6, also the entries in the Index to the OEIS under Pellian equation
  •  ??? indicates an A-number to be filled in, probably by creating a new sequence.
  • Obviously many more discriminants could be added to this list - please help! (I have checked that this list of quadratic forms is correct through (about) d=52 (for d a multiple of 4) and d = 21 (for d == 1 mod 4).) Also many of these entries need b-files.
  • In contrast to the positive definite case, here we are not interested in how many times a given indefinite form represents n (for here it is either 0 or infinity) except where the discriminant is a square and the number of solutions for a given n is finite.

The list

5. Computer programs

Remarks

  • Note that for most of these programs the goal is to find all nonnegative numbers, or all primes, represented by a binary quadratic form ax^2+bxy+cy^2 allowing any integer values of x and y. There are two exceptions:
    • Will Jagy's program Conway_Positive_Primitive finds the positive numbers that are primitively represented by an indefinite form.
    • The programs QuadPrimes and QuadPrimes2 look for primes that are represented by a positive definite form, allowing only nonnegative integer values of x and y.
  • Many of the programs currently in the OEIS for finding numbers represented by an indefinite form use brute force (see A031363, A035251, A084916, A089270, etc.). This is a notoriously unreliable method, and all these entries (especially any b-files) should be recomputed using Jagy's program Conway_Positive_All or Maple's isolve (See Section 5). (A031363 has now been corrected.)

PARI program fc for primes represented by a definite or indefinite binary quadratic form

  • fc(a,b,c,M) uses PARI's qfbsolve command to list all primes p represented by ax^2+bxy+cy^2 in the range 2 <= p <= prime(M).
  • The discriminant b^2-4ac should not be a square. (If it is, use the Maple program fb below.)
  • Works for both positive definite and indefinite forms.
fc(a,b,c,M,filename) = {
  if(filename,my(n=0),my(L=List()));
  forprime(p=2,prime(M),
    my(t = qfbsolve(Qfb(a,b,c),p));
    if(t, if(filename,write(filename,m++," ",p),listput(L,p)))
  );
  if(!filename,Vec(L))
};
fc(1,1,2,100) \\ Gives A045373

Maple program fd for numbers and primes represented by a positive definite binary quadratic form

  • The Maple program fd(a,b,c,M) will return two lists: all numbers in the range 0 <= n <= M represented by the positive definite binary quadratic form ax^2+bxy+cy^2, and all primes in that range that are represented.
fd:=proc(a,b,c,M) local dd,xlim,ylim,x,y,t1,t2,t3,t4,i;
dd:=4*a*c-b^2;
if dd<=0 then error "Form should be positive definite."; break; fi;
t1:={};
xlim:=ceil( sqrt(M/a)*(1+abs(b)/sqrt(dd)));
ylim:=ceil( 2*sqrt(a*M/dd));
for x from 0 to xlim do
for y from -ylim to ylim do
t2 := a*x^2+b*x*y+c*y^2;
if t2 <= M then t1:={op(t1),t2}; fi; od: od:
t3:=sort(convert(t1,list));
t4:=[];
for i from 1 to nops(t3) do
   if isprime(t3[i]) then t4:=[op(t4),t3[i]]; fi; od:
[[seq(t3[i],i=1..nops(t3))], [seq(t4[i],i=1..nops(t4))]];
end;
fd(1,1,3,500); # Gives A028954 and A056874
 

Will Jagy's programs for listing nonnegative numbers represented by an indefinite binary quadratic form

  • There are two programs, written in C++ (also available in Sage and Maple, see below), and very fast.
  • The program Conway_Positive_All.cc finds all positive integers n represented by the indefinite form ax^+bxy+cy^2 with n <= M.
    • The source file can be found here
    • To compile it, type g++ -o Conway_Positive_All Conway_Positive_All.cc -lm
    • To use it, type for example ./Conway_Positive_All 1 0 -3 100 (which produces A084916)
    • A sample output can be found here
  • The program Conway_Positive_Primitive.cc finds all positive integers n that are primitively represented by the indefinite form ax^+bxy+cy^2 with n <= M.
    • The source file can be found here
    • To compile it, type g++ -o Conway_Positive_Primitive Conway_Positive_Primitive.cc -lm
    • To use it, type for example ./Conway_Positive_Primitive 1 0 -3 100 (which produces A243655)
    • A sample output can be found here
  • Peter Luschny has ported these programs to Sage and Maple: see here.
  • Michael Somos has ported Conway_Positive_Primitive.cc to PARI/GP - see here

Using Maple's isolve command for listing nonnegative numbers represented by an indefinite binary quadratic form

  • Maple's isolve command can be used to do this. Here is a program from Robert Israel for A031363:

select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]);

If the discriminant is a square: Maple program for numbers and primes represented

  • If the discriminant is a nonzero perfect square (which implies that the quadric form is indefinite), the Maple program fc(a,b,c,M) will return two lists: all numbers in the range 0 <= n <= M represented by the form ax^2+bxy+cy^2, and all primes in that range that are represented.
fb:=proc(a,b,c,M) local s,t1,t2,n,d,dp;
if not issqr(b^2-4*a*c) then error "discriminant not a square"; return; fi;
s:=sqrt(b^2-4*a*c); t1:={0}; t2:={};
for n from 1 to M do
for d in numtheory[divisors](4*a*n) do dp:=4*a*n/d;
if ((d-dp) mod 2*s) = 0 and (((b+s)*dp-(b-s)*d) mod 4*a*s) = 0
then t1:={op(t1),n}; if isprime(n) then t2:={op(t2),n}; fi; break; fi;
                                    od:
                  od:
[sort(convert(t1,list)), sort(convert(t2,list))];
end;
fb(1,1,-2,500); # Gives A242660 and A002476
 

Mathematica program for listing nonnegative numbers represented by an indefinite binary quadratic form

  • Enter the coefficients by hand. For example, to find the nonnegative numbers represented by 2x^2+3xy-4y^2 (A035269), run:
Reap[For[n = 0, n <= 100, n++,  If[FindInstance[2*x^2 + 3*x*y - 4*y^2 == n,
{x, y}, Integers, 1] != {}, Sow[n]]]][[2, 1]]
 
  • In many cases this program is so slow as to be unusable. Use Jagy's programs instead.
  • To find the primes in this sequence, assuming the command worked, follow the above command by this:
Select[%,PrimeQ]
 

The defective Mathematica program QuadPrimes

QuadPrimes[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax}, d = b^2 - 4a*c; 
If[a > 0 && c > 0 && d < 0, xMax = Sqrt[lmt/a]]; 
Do[yMax = ((-b)*x + Sqrt[4c*lmt + d*x^2])/(2c); Do[p = a*x^2 + b*x*y + c*y^2; 
If[ PrimeQ[ p]  && !MemberQ[ lst, p], AppendTo[ lst, p]], {y, 0, yMax}], {x, 0, xMax}]; 
Sort[ lst]];
QuadPrimes[1, 1, 2, 1000] (* Should give A106856 *)
  
  • If b is nonzero, this program can give wrong answers, since xMax should be sqrt(lmt/a)*(1+|b|/sqrt(-d)), as one sees by completing the square. This only makes a difference for primes close to the limit.
  • For a specific example where QuadPrimes gives wrong answers, run QuadPrimes[2, -1, 17, 100000]. This finds 858 primes, but because the range of x is too small, it misses the primes 99713 and 99833, as can be seen by running QuadPrimes[2, -1, 17, 100500].
  • See Section 7 for details of the sequences where b-files computed using QuadPrimes were found to be in error.

A corrected version, QuadPrimes2

  • The following Mathematica program has the correct limits.
QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax},
   d = b^2 - 4 a*c;
   If[a > 0 && c > 0 && d < 0,
    xMax = Sqrt[lmt/a]*(1 + Abs[b]/Floor[Sqrt[-d]]);
    Do[
     If[4 c*lmt + d*x^2 >= 0, 
      yMax = ((-b)*x + Sqrt[4 c*lmt + d*x^2])/(2 c), yMax = 0];
     Do[
      p = a*x^2 + b*x*y + c*y^2;
      If[PrimeQ[p] && p <= lmt && ! MemberQ[lst, p], AppendTo[lst, p]];
      , {y, 0, yMax}];
     , {x, 0, xMax}];
    ,
    Print["Error, incorrect form: {a,b,c}=", {a, b, c}, ", d=", d];
    ];
   Sort[lst]];
QuadPrimes2[1, 1, 2, 1000]  (* Gives A106856 *)
 
  • The following alternative version is better for creating a b-file.
QuadPrimes2[a_, b_, c_, lmt_] := Module[{p, d, lst = {}, xMax, yMax},
   d = b^2 - 4 a*c;
   If[a > 0 && c > 0 && d < 0,
    xMax = Sqrt[lmt/a]*(1 + Abs[b]/Floor[Sqrt[-d]]);
    Do[
     If[4 c*lmt + d*x^2 >= 0, 
      yMax = ((-b)*x + Sqrt[4 c*lmt + d*x^2])/(2 c), yMax = 0];
     Do[
      p = a*x^2 + b*x*y + c*y^2;
      If[PrimeQ[p] && p <= lmt && ! MemberQ[lst, p], AppendTo[lst, p]];
      , {y, 0, yMax}];
     , {x, 0, xMax}];
    ,
    Print["Error, incorrect form: {a,b,c}=", {a, b, c}, ", d=", d];
    ];
   Sort[lst]];

t2 = QuadPrimes2[1, 1, 2, 350000];
Length[t2] (* See how many primes were found *)
t2[[Length[t2]]] (* Look at last term *)
For[n=1, n <= 10000, n++, Print[n, " ", t2[[n]]]] (* Create b-file *)
 

MAGMA commands

  • MAGMA has a number of commands for studying binary quadratic forms.
  • The commands Q := BinaryQuadraticForms(d); ReducedForms(Q); produce a list of the reduced forms of discriminant d, where d can be negative or positive.
  • To find the numbers represented by a positive definite quadratic form one can say Q := BinaryQuadraticForms(d); f := Q![a,b,c]; T := ThetaSeries(f, 100);, which gives the numbers represented (the exponents) and the numbers of representations (the coefficients). I do not know if MAGMA can find the numbers represented by an indefinite form.

PARI's command qfsolve

  • This will be included in PARI 2.8, and will find a rational solution to ax^2+bxy+cy^2 = n.

Dario Alpern's On-Line Quadratic Equation Solver

Peter Luschny's programs

  • Peter Luschny has a satellite page where he has ported some of these programs to Sage and Maple: see here.

6. References on binary quadratic forms

Remarks

  • For the general theory, see especially the books by Buell, Cox, and Zagier.
  • The papers by Delone, Kitaoka, Schering, Timofeev, and Watson (and Estes-Lee) are concerned with the question of when two positive definite binary quadratic forms represent the same numbers.
  • The two papers by Li are concerned with the question of when two indefinite binary quadratic forms represent the same numbers.
  • The papers by Jagy-Kaplansky and Voight are concerned with the question of when two binary quadratic forms represent the same primes.

References

  • Z. I. Borevich and I. R. Shafarevich, Number Theory, Translated from the Russian by Newcomb Greenleaf, Academic Press, New York, 1966.
  • D. A. Buell, Binary Quadratic Forms, Springer-Verlag, NY, 1989.
  • G. Chrystal, Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges, 7th. ed., Chelsea, NY, 1964. See especially pp. 478-490.
  • Harvey Cohn, A classical invitation to algebraic numbers and class fields. Universitext. Springer-Verlag, New York-Heidelberg, 1978. xiii+328 pp. ISBN: 0-387-90345-3 MR0506156 (80c:12001). See especially Chapter 14.
  • J. H. Conway, The Sensual (Quadratic) Form, Volume 26 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1997.
  • J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, NY, 3rd. ed., 1999. See especially Chap. 15 and the tables on pages 360 and 362-363.
  • David A. Cox, Primes of the Form x^2 + n y^2, Wiley, 1989; 2nd. ed. 2013.
  • H. M. Edwards, Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory, Springer-Verlag, NY, 1977. See especially Chapter 8.
  • B. N. Delone. Geometry of positive quadratic forms, addendum, Uspechi Mat. Sri, 4 (1938), 102-164.
  • Estes, Dennis R.; Lee, Albert. Classifying rank 2 quadratic lattices by representations. Nova J. Algebra Geom. 2 (1993), no. 2, 201--217. MR1273739 (95d:11041)
  • G. Frei, Euler's convenient numbers, Math. Intelligencer, Vol. 7, No. 3 (1985), 55-58 and 64.
  • C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965.
  • E. L. Ince, Cycles of Reduced Ideals in Quadratic Fields. British Association Mathematical Tables, Vol. 4, London, 1934.
  • Y. Kitaoka, On the relation between the positive-definite quadratic forms with the same representation numbers, Proc. Jap. Acad., 47 (1971), 439-441.
  • De Lang Li, Indefinite binary forms representing the same numbers, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 1, 29-33.
  • De Lang Li, Representation of numbers by binary quadratic forms, Acta Math. Sinica (N.S.) 3 (1987), no. 1, 58-65.
  • M. Schering. Theoremes relatifs aux formes quadratiques qui representent les memes nombres, J. Math, pures el appl., 2 serie 4 (1859).
  • V. N. Timofeev. On positive quadratic forms, representing the same numbers, Uspekhi Mat. Nauk, 18 (1963), 191-193.
  • Watson, G. L. Determination of a binary quadratic form by its values at integer points. Mathematika 26 (1979), no. 1, 72--75. MR0557128 (81e:10019).
  • Watson, G. L. Acknowledgement: "Determination of a binary quadratic form by its values at integer points [Mathematika 26 (1979), no. 1, 72-75; MR 81e:10019]. Mathematika 27 (1980), no. 2, 188 (1981). MR0610704 (82d:10037)
  • D. B. Zagier, Zetafunktionen und Quadratische Körper, Springer-Verlag, Berlin, 1981.

Links

7. List of sequences from positive definite forms that need to be checked

  • This section gives the full list of sequences based on positive definite quadratic forms ax^2+bxy+cy^2 with b != 0 and which mention the defective program QuadPrimes, and therefore need to be checked.
  • See Section 5 for a description of the bug in QuadPrimes and for replacement programs.
  • Two replacement programs for QuadPrimes are QuadPrimes2 (if one wishes to reproduce what QuadPrimes claims to do), or the faster PARI program fc if you simply want all the primes represented by a binary quadratic form.
  • Note that both the Data lines and (if present) the b-file in the OEIS entry need to be checked.
  • Once you have checked - or corrected - any of these sequences, add a comment in Section 7 and in the OEIS entry, saying something like "Checked by XXX, June 15, 2014 using the program YYY". Then delete the defective program QuadPrimes from the entry in the OEIS.
  • All of the sequences listed below have now been checked. Eleven b-files were found to have errors and were corrected as is noted in the comments below. - Ray Chandler, Aug 05 2014.
  • Sequences to be checked:
    • A056874 QuadPrimes[1, -1, 3, 10000] [Checked by NJAS, Jun 16 using PARI program fc]
    • A106856 QuadPrimes[1, 1, 2, 1000] [Checked by NJAS Jun 06 2014 using QuadPrimes2]
    • A106857 QuadPrimes[1, 1, 3, 10000] [Checked by NJAS Jun 16 2014 using QuadPrimes2]
    • A106858 QuadPrimes[2, 1, 2, 10000] [Checked by NJAS, Jun 17 2014 using QuadPrimes2]
    • A106859 Union[QuadPrimes[2, 1, 2, 10000], QuadPrimes[2, -1, 2, 10000]] Checked by NJAS Jun 06 2014 using PARI program fc]
    • A106860 QuadPrimes[2, -1, 2, 1000000] [Checked by Rick L. Shepherd 13:25, 25 July 2014 (UTC) using QuadPrimes2]
    • A106861 QuadPrimes[1, 1, 4, 1000000] [Checked by Rick L. Shepherd 14:02, 25 July 2014 (UTC) using QuadPrimes2]
    • A106862 QuadPrimes[1, 1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106863 QuadPrimes[1, -1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106864 QuadPrimes[2, 2, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106865 QuadPrimes[2, -2, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106866 QuadPrimes[2, 1, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106867 Union[QuadPrimes[2, 1, 3, 10000], QuadPrimes[2, -1, 3, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106868 QuadPrimes[2, -1, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106869 QuadPrimes[1, 1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106870 QuadPrimes[1, 1, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106871 QuadPrimes[2, 1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106872 Union[QuadPrimes[2, 1, 4, 10000], QuadPrimes[2, -1, 4, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106873 QuadPrimes[2, -1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106874 QuadPrimes[1, 1, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106875 QuadPrimes[3, 2, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106876 QuadPrimes[3, -2, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106877 QuadPrimes[3, 1, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106878 Union[QuadPrimes[3, 1, 3, 10000], QuadPrimes[3, -1, 3, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106879 QuadPrimes[3, -1, 3, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106880 QuadPrimes[1, 1, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106881 QuadPrimes[1, -1, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106882 QuadPrimes[2, 2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106883 QuadPrimes[3, 3, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106884 QuadPrimes[3, -3, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106885 QuadPrimes[2, 1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106886 Union[QuadPrimes[2, 1, 5, 10000], QuadPrimes[2, -1, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106887 QuadPrimes[2, -1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106888 QuadPrimes[1, 1, 10, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106890 QuadPrimes[1, 1, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106891 QuadPrimes[1, -1, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106892 QuadPrimes[3, 2, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106893 QuadPrimes[3, -2, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106894 QuadPrimes[3, 1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106895 Union[QuadPrimes[3, 1, 4, 10000], QuadPrimes[3, -1, 4, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106896 QuadPrimes[3, -1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106897 QuadPrimes[2, 1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106898 Union[QuadPrimes[2, 1, 6, 10000], QuadPrimes[2, -1, 6, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106899 QuadPrimes[2, -1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106900 QuadPrimes[1, 1, 12, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106901 QuadPrimes[3, 3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106902 QuadPrimes[3, -3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106903 QuadPrimes[1, 1, 13, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106904 QuadPrimes[1, -1, 13, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106905 QuadPrimes[2, 2, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106906 QuadPrimes[2, -2, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106907 QuadPrimes[4, 3, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106908 Union[QuadPrimes[4, 3, 4, 10000], QuadPrimes[4, -3, 4, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106909 QuadPrimes[4, -3, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106910 QuadPrimes[2, 1, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106911 Union[QuadPrimes[2, 1, 7, 10000], QuadPrimes[2, -1, 7, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106912 QuadPrimes[2, -1, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106913 QuadPrimes[1, 1, 14, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106914 QuadPrimes[3, 2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106915 Union[QuadPrimes[3, 2, 5, 10000], QuadPrimes[3, -2, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106916 QuadPrimes[3, -2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106918 QuadPrimes[3, 1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106919 Union[QuadPrimes[3, 1, 5, 10000], QuadPrimes[3, -1, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106920 QuadPrimes[3, -1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106921 QuadPrimes[1, 1, 15, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106922 QuadPrimes[1, -1, 15, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106923 QuadPrimes[4, 1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106924 Union[QuadPrimes[4, 1, 4, 10000], QuadPrimes[4, -1, 4, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106925 QuadPrimes[4, -1, 4, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106926 QuadPrimes[2, 1, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106927 Union[QuadPrimes[2, 1, 8, 10000], QuadPrimes[2, -1, 8, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106928 QuadPrimes[2, -1, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106929 QuadPrimes[1, 1, 16, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106930 QuadPrimes[1, -1, 16, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106931 QuadPrimes[4, 4, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106932 QuadPrimes[1, 1, 17, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106933 QuadPrimes[1, -1, 17, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106934 QuadPrimes[3, 2, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106935 Union[QuadPrimes[3, 2, 6, 10000], QuadPrimes[3, -2, 6, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106936 QuadPrimes[3, -2, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106937 QuadPrimes[2, 2, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106938 QuadPrimes[2, -2, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106939 QuadPrimes[4, 3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106940 Union[QuadPrimes[4, 3, 5, 10000], QuadPrimes[4, -3, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106941 QuadPrimes[4, -3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106942 QuadPrimes[3, 1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106943 Union[QuadPrimes[3, 1, 6, 10000], QuadPrimes[3, -1, 6, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106944 QuadPrimes[3, -1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106945 QuadPrimes[2, 1, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106946 Union[QuadPrimes[2, 1, 9, 10000], QuadPrimes[2, -1, 9, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106947 QuadPrimes[2, -1, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106948 QuadPrimes[1, 1, 18, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106951 QuadPrimes[3, 3, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106952 QuadPrimes[3, -3, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106953 QuadPrimes[4, 2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106954 Union[QuadPrimes[4, 2, 5, 10000], QuadPrimes[4, -2, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106955 QuadPrimes[4, -2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106956 QuadPrimes[4, 1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106957 Union[QuadPrimes[4, 1, 5, 10000], QuadPrimes[4, -1, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106958 QuadPrimes[4, -1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106959 QuadPrimes[2, 1, 10, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106960 Union[QuadPrimes[2, 1, 10, 10000], QuadPrimes[2, -1, 10, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106961 QuadPrimes[2, -1, 10, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106962 QuadPrimes[1, 1, 20, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106964 QuadPrimes[3, 2, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106965 QuadPrimes[3, -2, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106966 QuadPrimes[3, 1, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106967 Union[QuadPrimes[3, 1, 7, 10000], QuadPrimes[3, -1, 7, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106968 QuadPrimes[3, -1, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106969 QuadPrimes[1, 1, 21, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106970 QuadPrimes[1, -1, 21, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106971 QuadPrimes[5, 4, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106972 Union[QuadPrimes[5, 4, 5, 10000], QuadPrimes[5, -4, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106973 QuadPrimes[5, -4, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106974 QuadPrimes[2, 2, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106975 QuadPrimes[4, 3, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106976 Union[QuadPrimes[4, 3, 6, 10000], QuadPrimes[4, -3, 6, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106977 QuadPrimes[4, -3, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106978 QuadPrimes[3, 3, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106979 QuadPrimes[3, -3, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106980 QuadPrimes[2, 1, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106981 Union[QuadPrimes[2, 1, 11, 10000], QuadPrimes[2, -1, 11, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106982 QuadPrimes[2, -1, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106983 QuadPrimes[1, 1, 22, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106985 QuadPrimes[5, 3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106986 Union[QuadPrimes[5, 3, 5, 10000], QuadPrimes[5, -3, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106987 QuadPrimes[5, -3, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106988 QuadPrimes[1, 1, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106989 QuadPrimes[1, -1, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106990 QuadPrimes[5, 5, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106991 QuadPrimes[5, -5, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106992 QuadPrimes[4, 1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106993 Union[QuadPrimes[4, 1, 6, 10000], QuadPrimes[4, -1, 6, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106994 QuadPrimes[4, -1, 6, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106995 QuadPrimes[3, 1, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106996 Union[QuadPrimes[3, 1, 8, 10000], QuadPrimes[3, -1, 8, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106997 QuadPrimes[3, -1, 8, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106998 QuadPrimes[2, 1, 12, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A106999 Union[QuadPrimes[2, 1, 12, 10000], QuadPrimes[2, -1, 12, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107000 QuadPrimes[2, -1, 12, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107001 QuadPrimes[1, 1, 24, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107002 QuadPrimes[5, 2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107003 Union[QuadPrimes[5, 2, 5, 10000], QuadPrimes[5, -2, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107004 QuadPrimes[5, -2, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107005 QuadPrimes[4, 4, 7, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107006 QuadPrimes[4, -4, 7, 10000] [Checked by NJAS, Jun 08 2014]
    • A107009 QuadPrimes[5, 1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107010 Union[QuadPrimes[5, 1, 5, 10000], QuadPrimes[5, -1, 5, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107011 QuadPrimes[5, -1, 5, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107012 QuadPrimes[1, 1, 25, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A107013 QuadPrimes[1, -1, 25, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A121243 QuadPrimes[4, -4, 9, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139827 QuadPrimes[2, -2, 17, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139828 QuadPrimes[6, -6, 7, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139829 QuadPrimes[4, -4, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139830 Union[QuadPrimes[7, 6, 7, 10000], QuadPrimes[7, -6, 7, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139831 QuadPrimes[2, -2, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139832 Union[QuadPrimes[7, 4, 7, 10000], QuadPrimes[7, -4, 7, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139833 QuadPrimes[2, -2, 29, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139834 QuadPrimes[6, -6, 11, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139835 QuadPrimes[2, -2, 43, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139836 QuadPrimes[10, -10, 11, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139837 QuadPrimes[4, -4, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139838 QuadPrimes[8, -8, 13, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139839 QuadPrimes[2, -2, 47, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139840 QuadPrimes[6, -6, 17, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139844 QuadPrimes[2, -2, 53, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139847 QuadPrimes[6, -6, 19, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139849 QuadPrimes[10, -10, 13, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139850 Union[QuadPrimes[11, 8, 11, 10000], QuadPrimes[11, -8, 11, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139851 QuadPrimes[4, -4, 29, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139853 Union[QuadPrimes[11, 6, 11, 10000], QuadPrimes[11, -6, 11, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139855 QuadPrimes[4, -4, 31, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139858 QuadPrimes[8, -8, 17, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139859 Union[QuadPrimes[11, 2, 11, 10000], QuadPrimes[11, -2, 11, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139860 QuadPrimes[12, -12, 13, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139864 QuadPrimes[2, -2, 67, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139866 Union[QuadPrimes[13, 12, 13, 10000], QuadPrimes[13, -12, 13, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139867 QuadPrimes[2, -2, 83, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139870 QuadPrimes[6, -6, 29, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139871 QuadPrimes[10, -10, 19, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139873 Union[QuadPrimes[13, 4, 13, 10000], QuadPrimes[13, -4, 13, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139875 QuadPrimes[4, -4, 43, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139878 QuadPrimes[8, -8, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139879 QuadPrimes[12, -12, 17, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139880 Union[QuadPrimes[13, 2, 13, 10000], QuadPrimes[13, -2, 13, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139881 QuadPrimes[2, -2, 89, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139883 QuadPrimes[6, -6, 31, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139894 QuadPrimes[4, -4, 59, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139896 QuadPrimes[8, -8, 31, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139898 QuadPrimes[4, -4, 61, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139900 QuadPrimes[12, -12, 23, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139902 QuadPrimes[16, -16, 19, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139903 Union[QuadPrimes[17, 14, 17, 10000], QuadPrimes[17, -14, 17, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139904 QuadPrimes[2, -2, 127, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139906 Union[QuadPrimes[17, 12, 17, 10000], QuadPrimes[17, -12, 17, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139907 QuadPrimes[2, -2, 137, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139909 QuadPrimes[6, -6, 47, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139912 QuadPrimes[14, -14, 23, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139913 Union[QuadPrimes[17, 8, 17, 10000], QuadPrimes[17, -8, 17, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139914 QuadPrimes[4, -4, 71, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139918 QuadPrimes[8, -8, 37, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139919 Union[QuadPrimes[17, 6, 17, 10000], QuadPrimes[17, -6, 17, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139920 Union[QuadPrimes[19, 18, 19, 10000], QuadPrimes[19, -18, 19, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139922 QuadPrimes[4, -4, 79, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139924 QuadPrimes[8, -8, 41, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139925 QuadPrimes[12, -12, 29, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139927 Union[QuadPrimes[19, 14, 19, 10000], QuadPrimes[19, -14, 19, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139935 QuadPrimes[2, -2, 173, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139938 QuadPrimes[6, -6, 59, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139939 QuadPrimes[10, -10, 37, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139941 Union[QuadPrimes[19, 8, 19, 10000], QuadPrimes[19, -8, 19, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139942 QuadPrimes[2, -2, 179, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139944 QuadPrimes[6, -6, 61, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139946 QuadPrimes[14, -14, 29, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139948 Union[QuadPrimes[19, 4, 19, 10000], QuadPrimes[19, -4, 19, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139949 QuadPrimes[2, -2, 193, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139952 QuadPrimes[10, -10, 41, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139954 QuadPrimes[14, -14, 31, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139955 QuadPrimes[22, -22, 23, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139957 QuadPrimes[4, -4, 103, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139959 QuadPrimes[8, -8, 53, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139960 QuadPrimes[12, -12, 37, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139962 Union[QuadPrimes[23, 22, 23, 10000], QuadPrimes[23, -22, 23, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139970 QuadPrimes[4, -4, 131, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139973 QuadPrimes[8, -8, 67, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139975 QuadPrimes[20, -20, 31, 10000] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139976 Union[QuadPrimes[23, 6, 23, 10000], QuadPrimes[23, -6, 23, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139977 QuadPrimes[4, -4, 191, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139980 QuadPrimes[8, -8, 97, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139982 QuadPrimes[20, -20, 43, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139983 Union[QuadPrimes[29, 18, 29, 10000], QuadPrimes[29, -18, 29, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139985 QuadPrimes[4, -4, 211, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139989 QuadPrimes[8, -8, 107, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139990 QuadPrimes[12, -12, 73, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139992 QuadPrimes[20, -20, 47, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139995 QuadPrimes[24, -24, 41, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139996 QuadPrimes[28, -28, 37, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139997 Union[QuadPrimes[29, 2, 29, 10000], QuadPrimes[29, -2, 29, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A139998 Union[QuadPrimes[31, 22, 31, 10000], QuadPrimes[31, -22, 31, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140000 QuadPrimes[4, -4, 331, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140003 QuadPrimes[8, -8, 167, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140005 QuadPrimes[12, -12, 113, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140007 QuadPrimes[20, -20, 71, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140009 QuadPrimes[24, -24, 61, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140011 Union[QuadPrimes[37, 14, 37, 10000], QuadPrimes[37, -14, 37, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140012 QuadPrimes[40, -40, 43, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140013 Union[QuadPrimes[41, 38, 41, 10000], QuadPrimes[41, -38, 41, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140014 QuadPrimes[2, -2, 683, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140017 QuadPrimes[6, -6, 229, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140019 QuadPrimes[10, -10, 139, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140021 QuadPrimes[14, -14, 101, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140024 QuadPrimes[26, -26, 59, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140025 QuadPrimes[30, -30, 53, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140027 Union[QuadPrimes[37, 4, 37, 10000], QuadPrimes[37, -4, 37, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140028 QuadPrimes[42, -42, 43, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140030 QuadPrimes[4, -4, 463, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140033 QuadPrimes[8, -8, 233, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140035 QuadPrimes[12, -12, 157, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140038 QuadPrimes[24, -24, 83, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140039 QuadPrimes[28, -28, 73, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140041 Union[QuadPrimes[43, 2, 43, 10000], QuadPrimes[43, -2, 43, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140042 QuadPrimes[44, -44, 53, 10000] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140043 Union[QuadPrimes[47, 38, 47, 10000], QuadPrimes[47, -38, 47, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140613 Union[QuadPrimes[7, 6, 39, 10000], QuadPrimes[7, -6, 39, 10000]] [Checked by NJAS Jun 07 2014 using new PARI program added there]
    • A140614 Union[QuadPrimes[15, 12, 20, 10000], QuadPrimes[15, -12, 20, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140615 Union[QuadPrimes[13, 6, 21, 10000], QuadPrimes[13, -6, 21, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140616 Union[QuadPrimes[5, 4, 68, 10000], QuadPrimes[5, -4, 68, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140617 Union[QuadPrimes[11, 8, 32, 10000], QuadPrimes[11, -8, 32, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140618 Union[QuadPrimes[20, 4, 23, 10000], QuadPrimes[20, -4, 23, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140619 Union[QuadPrimes[19, 4, 28, 10000], QuadPrimes[19, -4, 28, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140620 Union[QuadPrimes[23, 4, 68, 10000], QuadPrimes[23, -4, 68, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140621 Union[QuadPrimes[28, 12, 57, 10000], QuadPrimes[28, -12, 57, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140622 Union[QuadPrimes[21, 12, 76, 10000], QuadPrimes[21, -12, 76, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140623 Union[QuadPrimes[35, 30, 51, 10000], QuadPrimes[35, -30, 51, 10000]] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140624 Union[QuadPrimes[19, 14, 91, 10000], QuadPrimes[19, -14, 91, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140625 Union[QuadPrimes[28, 20, 85, 10000], QuadPrimes[28, -20, 85, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140626 Union[QuadPrimes[51, 48, 56, 10000], QuadPrimes[51, -48, 56, 10000]] [Checked and corrected by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140627 Union[QuadPrimes[33, 24, 88, 10000], QuadPrimes[33, -24, 88, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140628 Union[QuadPrimes[39, 6, 71, 10000], QuadPrimes[39, -6, 71, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140629 Union[QuadPrimes[76, 20, 145, 10000], QuadPrimes[76, -20, 145, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140630 Union[QuadPrimes[88, 32, 127, 10000], QuadPrimes[88, -32, 127, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140631 Union[QuadPrimes[57, 18, 193, 10000], QuadPrimes[57, -18, 193, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140632 Union[QuadPrimes[55, 10, 199, 10000], QuadPrimes[55, -10, 199, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]
    • A140633 Union[QuadPrimes[7, 4, 52, 10000], QuadPrimes[7, -4, 52, 10000]] [Checked by Ray Chandler, Jul 26 2014 using QuadPrimes2]

8. Want to help?

  • Many of the programs currently in the OEIS for finding numbers represented by an indefinite form use brute force (see A031363, A035251, A084916, A089270, A243180, A243181, etc.). This is a notoriously unreliable method, and all these entries (especially any b-files) should be recomputed using Jagy's program Conway_Positive_All or Maple's isolve (See Section 5). A031363 has now been corrected.
  • Another important task is to check that the lists in Sections 3 and 4 are complete as far as they go, and to extend them by including further discriminants.
    • Pick a discriminant d, run MAGMA (see Section 5) to find all reduced forms of that discriminant, and for each form, check that the sequences of numbers represented, and primes represented, are in the OEIS and are listed in this web page.
  • Use Voight's article (see Section 6) to resolve the questions in Section 3 about whether two apparently identical sequences of primes really are the same.

9. About this page

  • Created by N. J. A. Sloane, June 05 - 12, 2014.
  • Thanks to Jean-François Alcover, Bill Allombert, Ray Chandler, Charles R Greathouse IV, Will Jagy, Peter Luschny, Michael Somos, and others for helpful emails on the subject of this page.
  • If you make any changes to this page, please sign and date them.