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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. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

REFERENCES

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

LINKS

Table of n, a(n) for n=1..70.

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}.

EXAMPLE

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

CROSSREFS

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

Sequence in context: A179460 A344281 A171886 * A018559 A057196 A080637

Adjacent sequences: A343235 A343236 A343237 * A343239 A343240 A343241

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, May 16 2021

STATUS

approved

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Last modified December 7 13:02 EST 2022. Contains 358656 sequences. (Running on oeis4.)