cp's OEIS Frontend

It is a classical result that p is of the form x^2 + 5y^2 if and only if p = 5 or p == 1 or 9 mod 20 (see Cox, page 33). - N. J. A. Sloane, Sep 20 2012

Except for 5, also primes of the form x^2 + 25y^2. See A140633. - T. D. Noe, May 19 2008

Or, 5 and all primes p that divide Fibonacci((p - 1)/2) = A121568(n). - Alexander Adamchuk, Aug 07 2006

This is one of the bisections of sequence A343238. The other sequence is A344231.

This is a proper subsequence of A029718.

The primes in this sequence are given in A106865.

See A344231 for more details.

The reduced form [2, 2, 3] represents the proper (determinant +1) equivalence class of one of the two genera (genus II) of discriminant -20. The multiplicative generic characters for discriminant Disc = -20 have values Jacobi(a(n)|5) = -1 and Jacobi(-1|a(n)) = -1, for odd a(n) not divisible by 5. See Buell, p. 52.

The product of any two odd a(n), not divisible by 5, is congruent to {1,5} (mod 8). See Buell, 4), p. 51.

For this genus II of Disc = -20 the positive integers represented are given by 2^a*5^b*Product_{j=1..PI} (pI_j)^(eI(j))*Product_{k=1..PII}(pII_k)^(eII(k)), with a and b from {0, 1}, but if PI = PII = 0 (empty products are 1) then (a, b) = (1, 0) or (1, 1), giving a(1) = 2 or a(4) = 10. The odd primes pI_j are from A033205 and the odd primes pII_j from the odd primes of A106865. The exponents of the second product satisfy: if a = 1 then PII >= 0, and if PII >=1 then Sum_{k=1..PII} eII(j) is even. If a = 0 then PII >= 1 and this sum is odd.

The neighboring numbers k (twins) begin: [42, 43], [82, 83], [122, 123] [162, 163], [202, 203], [282, 283], ...

For the solutions (X, Y) of F2 = [2, 2, 3] properly representing k = a(n) see A344234.

Discriminant = -20.