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A261246 Positive integers D such that the generalized Pell equation X^2 - D Y^2 = 2 is soluble. 8
2, 7, 14, 23, 31, 34, 46, 47, 62, 71, 79, 94, 98, 103, 119, 127, 142, 151, 158, 167, 191, 194, 199, 206, 223, 238, 239, 254, 263, 271, 287, 302, 311, 322, 334, 343, 359, 367, 382, 383, 386, 391, 398, 431, 439, 446, 463, 478, 479, 482, 487, 503, 511 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

For the fundamental positive solution x(n)^2 - a(n)*y(n)^2 = 2 see (x(n) = A261247(n), y(n) = A261248(n)), for n >= 1.

Conjecture: The sequence consists of all numbers D not a square and even D = 2*d has odd d with prime factors of the form 1 or 7 (mod 8). Odd D has prime factors of the form 1 or 7 (mod 8) but there is an odd number of primes of the form 7 (mod 8). The following will prove that these conditions for D are necessary in order to have solutions.

This conjecture is false. For the odd D case see the counterexamples in A263010, and for the even D in A264352. - Wolfdieter Lang, Nov 12 2015

If there is a solution for D, D not a square, then only one class of solution exists due to Nagell's Theorem 110, p. 208, because then 2 divides 2*D. All solutions will be proper because 2 is a prime.

For the even prime D = p = 2 the positive fundamental solution is [x(1) = 2, y(1) = 1].

For odd primes D = p there can be solutions only for p == +7 (mod 8), that is p from A007522. Then x and y are both odd. Proof: Consider a solution of x^2 - p*y^2 = 2. The parities of x and y have to be either even and even or odd and odd. For odd x one has x^2  == +1 (mod 8) (because x^2  = 8*T(X) + 1 with x = 2*X+1 and the triangular numbers T = A000217); similarly for y^2 if y is odd. In the even-even case x^2 and y^2 are both congruent to 4 (mod 8). The even-even case leads to 4 - 4*p = 2 (mod 8), excluding all odd p, namely p == 1, 3, 5, 7 (mod 8). The odd-odd case is 1 - p*1 = 2 (mod 8), and p == 1, 3, 5 (mod 8) are excluded. Therefore, only p == 7 (mod 8) qualifies for a solution, and then x and y will be both odd.

For D = p == 7 (mod 8) from A007522 one can test if there exists a fundamental positive solution (at most one class can exist, therefore there is either no solution or just one) [2*U(p)+1, 2*V(p)+1] by checking the two inequalities (see Nagell, eq. (4) and (5), p. 206) 0 <= V(p) < floor((Y(p)/sqrt(X(p) + 1) - 1)/2) and 0 <= U(p) <= floor((sqrt(X(p) + 1) - 1)/2), with the positive fundamental solution [X(p), Y(p)] of X^2 - p*Y^2 = +1. These solutions can be found in (A033313(k), A033317(k)) if A000037(k) is the prime p == 7 (mod 8) one is testing.

For composite even D there are solutions only if D/2 is odd. Proof: If D is even then x has to be even, hence x^2 == 0 (mod 4) and then D*y^2  == -2 (mod 4), hence D cannot be 0 (mod 4). Thus an even D can only be of the form D = 2*d with d odd. The modulo 3 and modulo 5 argument used in the next case will show that d can have only prime factors of the form +1 or -1 (mod 8).

For composite odd D one finds like above that the even-even x and y case is excluded, and the odd-odd case needs D == -1 (mod 8) == 7 (mod 8). Hence a candidate for D is from  A004771 - A007522. D cannot have any prime factor p of the form 3 or 5 (mod 8) because otherwise x^2 == 2 (mod p), but the Legendre symbol (2/p) = -1 for such p's (see, e.g., Nagell, Theorem 81, p. 136). For example, D = 15 = 3*5 cannot have a solution. Thus the only candidates for D have  prime factors p of the form +1 or +7 (mod 8), with the number of the latter ones being odd. E.g., D = 7*17 = 119 qualifies as a candidate and it has indeed solutions, namely the ones obtainable from the fundamental one [11, 1].

The general proper positive solutions for D(n) = a(n) are obtained from the fundamental ones [x(n), y(n)] given in A261247 and A261248 with the help of powers of the matrix M(n) = [[u(n), D(n)*v(n)], [v(n), u(n)]], where u(n) and v(n) are the positive fundamental solutions of U(n) - D(n)*V(n) = 1, by (x(n; k), y(n; k))^T = M(n)^k (x(n), y(n))^T (T for transposed), for k >= 0. [u(n), v(n)] = [A033313(j(n)), A033317(j(n))] if  A000037(j(n)) = D(n) = a(n).

REFERENCES

J. W. S. Cassels, Rational Quadratic Forms, Cambridge, 1978; see Chap. 3.

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964.

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..1000

Wolfdieter Lang, Binary Quadratic Forms (indefinite case).

EXAMPLE

The first fundamental solutions [x(n), y(n)] are (the first entry gives D(n)=a(n)):

[2, [2, 1]], [7, [3, 1]], [14, [4, 1]],

[23, [5, 1]], [31, [39, 7]], [34, [6, 1]],

[46, [156, 23]], [47, [7, 1]], [62, [8, 1]],

[71, [59, 7]], [79, [9, 1]], [94, [1464, 151]],

[98, [10, 1]], [103, [477, 47]], [119, [11, 1]],

[127, [2175, 193]], [142, [12, 1]],

[151, [41571, 3383]], [158, [88, 7]],

[167, [13, 1]], [191, [2999, 217]],

[194, [14, 1]], [199, [127539, 9041]],

[206, [244, 17]], [223, [15, 1]], [238, [108, 7]],

[239, [2489, 161]], ...

MATHEMATICA

Select[Range[600], False =!= Reduce[x^2 - # y^2 == 2, {x, y}, Integers] &] (* Giovanni Resta, Aug 12 2017 *)

CROSSREFS

Cf. A261247, A261248, A007522, A004771, A000217, A000037, A033313, A033317, A263010.

See also A038873 (2 and primes == +-1 mod 8), A001132.

Sequence in context: A114346 A325159 A087324 * A008865 A227582 A249852

Adjacent sequences:  A261243 A261244 A261245 * A261247 A261248 A261249

KEYWORD

nonn

AUTHOR

Wolfdieter Lang, Sep 06 2015

STATUS

approved

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Last modified July 20 11:28 EDT 2019. Contains 325180 sequences. (Running on oeis4.)