A343238 All positive integer moduli a(n) for which the congruence x^2 == -5 (mod a(n)) is solvable for integer x (representatives from {0, 1, ..., a(n)-1}); ordered increasingly.

1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 23, 27, 29, 30, 35, 41, 42, 43, 45, 46, 47, 49, 54, 58, 61, 63, 67, 69, 70, 81, 82, 83, 86, 87, 89, 90, 94, 98, 101, 103, 105, 107, 109, 115, 122, 123, 126, 127, 129, 134, 135, 138, 141, 145, 147, 149, 161, 162, 163, 166, 167, 174, 178, 181, 183, 189, 201, 202

Offset: 1

Author

Wolfdieter Lang, May 16 2021

Keywords

• nonn
• easy

Comments

This sequence includes A139513, that is, Legendre(-5, p) = +1 for odd primes not 5, that is, primes congruent to {1, 3, 7, 9} mod 20. Here 5 is a member of the sequence with solution x = 0.

The primes of this sequence are given in A240920.

The present sequence gives the numbers of the form 2^a*5^b*Product_{j=1..m} (p_j)^e(j), with a and b from {0, 1}, p_j a prime from {1, 3, 7, 9} (mod 20), i.e., from A139513, m >= 0 and e(j) >= 0 (this includes the number 1). These numbers are ordered increasingly.

This follows from the Legendre-symbol(-5, p)= +1 and the lifting theorem (see, e.g., Apostol, Theorem 5.30, p. 121-2) for p = 2 and 5 (no lifting for the solutions for p = 2 and p = 5), and the unique lifting for the primes satisfying Legendre-symbol(-5, p) = +1.

Therefore the number of representative solutions x from {0, 1, ..., a(n)-1}, denoted by M(a(n)), is 1 for precisely four cases: a(1) = 1 (x = 0), a(2) = 2 (x = 1), a(4) = 5 (x = 0) and a(8) = 10 = 2*5 (x = 5). For each of the mentioned prime powers there are just 2 solutions. This implies that for the number of solutions in the general a(n) case, n not 1, 2, 4, 8, only the primes p_j are of interest: M(a(n)) = 2^m(n).

For these solutions x see A343239, and for the multiplicity M(a(n)) see A343240.

This congruence is needed to find all proper solutions of the positive definite binary quadratic form of discriminant Disc = -20 = -4*5 representing k = a(n). The solutions x lead to the so-called representative parallel primitive forms (rpapfs). See A344231 for more details.

For a bisection see A344231 and A344232, related to integer solutions of X^2 + 5*Y^2 = A344231(k) and 2*X^2 + 2*X*Y + 3*Y^2 = A344232(k).

Examples

			a(3) = 3: two solutions 1 and 2.
a(7) = 3^2 = 9: two solutions 2 and 7.
a(8) = 10 = 2*5 only one solution 5.
a(53) = 135 = 5*3^3: two solutions 20 and 115.
		

References

• Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp 121, 122.

Crossrefs

Cf. A139513, A240920, A343239, A343240, A344231, A344232.

Programs

PARI

isok(k) = issquare(Mod(-5, k)); \\ Michel Marcus, Sep 17 2023

Formula

There exists at least one x from {0, 1, ..., m-1} satisfying x^2 + 5 == 0 (mod m), for positive integer m. These m values are then ordered increasingly as (a(n))_{n>=1}.