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A387129 Second numbers B = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and A = A387128(n).

%I A387129 #12 Aug 22 2025 00:20:56
%S A387129 1,3,1,5,3,7,9,3,9,11,1,9,7,15,3,11,17,13,3,19,9,21,15,21,7,19,23,15,
%T A387129 11,3,13,25,9,21,23,17,25,19,27,29,1,5,15,27,21,13,33,31,21,17,33,3,
%U A387129 15,29,27,33,27,1,5,21,39,37,39,11,39,13,33,41,29,17,27,33,43,15,3,27,23,9
%N A387129 Second numbers B = a(n) of two numbers (A, B) such that the sums 2*A^2 + B^2 = p == 3 mod 8, where p = A007520(n) and A = A387128(n).
%C A387129 Prime numbers p congruent to 3 mod 8 can be written as the sum of twice the square of an integer A and of the square of another integer B, i.e., 2*A^2 + B^2 = p, where A = A387128(n), B = a(n) (this sequence), and p = A007520(n) == 3 mod 8.
%C A387129 This representation is unique, i.e., for a given n, there are no other integer values of A(n) and B(n) such that p(n) = 2 * A(n)^2 + B(n)^2 where p(n) = A007520(n), the 3 mod 8 prime numbers.
%C A387129 For all n, A = A387128(n) and B = a(n) are always odd.
%C A387129 Terms are ordered according to increasing order of A007520(n).
%D A387129 Cartier P. "An Introduction to Zeta Functions", Chap 1.2, in eds. M. Waldschmidt, P. Moussa, J.M., Luck, C. Itzykson “From Number Theory to Physics”, Springer-Verlag, Berlin, pp. 22-41, 1960.
%D A387129 Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.
%D A387129 Hardy, G. H. and Wright, E. M. "Primes in k(i)" and "The Fundamental Theorem of Arithmetic in k(i)." 12.7 and 12.8 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 183-187, 1979.
%D A387129 Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.
%H A387129 Vladimir Pletser, <a href="/A387129/b387129.txt">Table of n, a(n) for n = 1..10000</a>
%F A387129 2 * A387128(n)^2 + a(n)^2 = A007520(n).
%e A387129 1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
%e A387129 21 belongs to the sequence as 2 * 5^2 + 21^2 = 491.
%Y A387129 Cf. A007520, A387128.
%K A387129 nonn,new
%O A387129 1,2
%A A387129 _Vladimir Pletser_, Aug 17 2025