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A387128 First numbers A = 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 B = A387129(n).

%I A387128 #12 Aug 22 2025 00:21:03
%S A387128 1,1,3,3,5,3,1,7,5,3,9,7,9,1,11,9,3,9,13,3,13,1,11,5,15,9,3,13,15,17,
%T A387128 15,3,17,11,9,15,9,15,7,3,21,21,19,11,17,21,1,9,19,21,7,25,23,15,17,
%U A387128 13,19,27,27,23,1,9,5,27,7,27,17,3,21,27,23,19,3,29,31,25,27,31,9,1,27
%N A387128 First numbers A = 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 B = A387129(n).
%C A387128 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 = a(n), B = A387129(n), and p = A007520(n) == 3 mod 8.
%C A387128 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 A387128 For all n, A = a(n) and B = A387129(n) are always odd.
%C A387128 Terms are ordered according to increasing order of A007520(n).
%D A387128 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 A387128 Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.
%D A387128 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 A387128 Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.
%H A387128 Vladimir Pletser, <a href="/A387128/b387128.txt">Table of n, a(n) for n = 1..10000</a>
%F A387128 2 * a(n)^2 + A387129(n)^2 = A007520(n).
%e A387128 1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
%e A387128 5 belongs to the sequence as 2 * 5^2 + 21^2 = 491.
%Y A387128 Cf. A007520, A387129.
%K A387128 nonn,new
%O A387128 1,3
%A A387128 _Vladimir Pletser_, Aug 17 2025