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A161681 Primes that are the difference between a cube and a square (conjectured values). 3
2, 7, 11, 13, 19, 23, 47, 53, 61, 67, 71, 79, 83, 89, 107, 109, 127, 139, 151, 167, 191, 193, 199, 223, 233, 239, 251, 271, 277, 293, 307, 359, 431, 433, 439, 463, 487, 499, 503, 547, 557, 587, 593, 599, 631, 647, 673, 683, 719, 727, 769, 797, 859, 887, 919 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
The primes found among the differences are sorted in ascending order and unique primes are then extracted. I call this a "conjectured" sequence since I cannot prove that somewhere on the road to infinity there will never exist an integer pair x,y such that x^3-y^2 = 3,5,17,..., missing prime. For example, testing x^3-y^2 for x,y up to 10000, the count of some duplicates are:
duplicate,count
7,2
11,2
47,3
431,7
503,7
1999,5
28279,11
Yet for 3,5,17,29,... I did not find even one.
[Comment from Charles R Greathouse IV, Nov 03 2009: 587 = 783^3 - 21910^2, 769 = 1025^3 - 32816^2, and 971 = 1295^3 - 46602^2 were skipped in the original.]
Conjecture: The number of primes in x^3-y*2 is infinite.
Conjecture: The number of duplicates for a given prime is finite. Then there is the other side - the primes that are not in the sequence 3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, ... Looks like a lot of twin components here. Do these have an analytical form? Is there such a thing as a undecidable sequence?
Range of A167224. - Reinhard Zumkeller, Oct 31 2009
LINKS
R. Zumkeller, Some Examples [From Reinhard Zumkeller, Oct 31 2009]
FORMULA
Integers x,y such that x^3-y^2 = p where p is prime. The generation bound is 10000.
EXAMPLE
3^3 - 4^2 = 15^3 - 58^2 = 11.
PROG
(PARI) diffcubesq(n) =
{
local(a, c=0, c2=0, j, k, y);
a=vector(floor(n^2/log(n^2)));
for(j=1, n,
for(k=1, n,
y=j^3-k^2;
if(ispseudoprime(y),
c++;
a[c]=y;
)
)
);
a=vecsort(a);
for(j=2, c/2,
if(a[j]!=a[j-1],
c2++;
print1(a[j]", ");
if(c2>100, break);
)
);
}
CROSSREFS
Cf. A000040.
Sequence in context: A138889 A097143 A038897 * A020583 A140557 A027697
KEYWORD
nonn
AUTHOR
Cino Hilliard, Jun 16 2009
EXTENSIONS
Extended and edited by Charles R Greathouse IV, Nov 03 2009
Further edits by N. J. A. Sloane, Nov 09 2009
STATUS
approved

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Last modified June 21 19:01 EDT 2024. Contains 373557 sequences. (Running on oeis4.)