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).

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, 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, 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

Offset

1,3

Comments

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.

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.

For all n, A = a(n) and B = A387129(n) are always odd.

Terms are ordered according to increasing order of A007520(n).

References

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.

Conway J.H. and Guy R.K. "The Book of Numbers", Chap. 5, Springer-Verlag, New York, pp. 127-149, 1996.

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.

Sierpinski W. "Elementary Theory of Numbers", Chap. 13.3 and 13.4, ed. A Schinzel, North Holland, Amsterdam, pp. 459-462, 1988.

Links

Vladimir Pletser, Table of n, a(n) for n = 1..10000

Formula

2 * a(n)^2 + A387129(n)^2 = A007520(n).

Example

1 belongs to the sequence as 2 * 1^2 + 1^2 = 3.
5 belongs to the sequence as 2 * 5^2 + 21^2 = 491.

Crossrefs

Cf. A007520, A387129.

Sequence in context: A115155 A136549 A302141 * A077924 A003569 A333596

Adjacent sequences: A387125 A387126 A387127 * A387129 A387130 A387131

Keyword

nonn

Author

Vladimir Pletser, Aug 17 2025

Status

approved