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A347831
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The positive integer moduli a(n) for which the congruence x*(x + 1) == -4 (mod a(n)) is solvable for integer x (representatives from {0, 1, ..., a(n)-1}); in increasing order.
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2
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1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 19, 20, 23, 24, 30, 31, 32, 34, 38, 40, 46, 47, 48, 51, 53, 57, 60, 61, 62, 64, 68, 69, 76, 79, 80, 83, 85, 92, 93, 94, 95, 96, 102, 106, 107, 109, 113, 114, 115, 120, 122, 124, 128, 136, 137, 138, 139, 141, 151, 152, 155, 158, 159, 160, 166, 167, 170, 173, 181
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OFFSET
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1,2
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COMMENTS
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The primes are 3, 5 and those given in A191018 (Jacobi(prime|15) = +1).
For a(n) neither a multiple of 3 nor of 5 the Jacobi(a(n)|15) = +1.
The sequence (a(n))_{n >=1} is the set S := {3^a*5^b*Product_{j=1..m} (p_j)^{e(j)}}, in increasing order, with a and b from {0, 1}, primes p_j from A191018, m >= 0, and the exponents e(j) >= 0. If a = b = m = 0 then S = {1} and a(1) = 1.
The multiplicity of the (representative) solutions x is 2^m(n) for modulus a(n) from the set S. Thus it is 1 for 1, 3, 5, and a power of 2 with m(n) >= 1. This follows from Jacobi(prime|15) = +1, and the lifting theorem for powers of these primes (see e.g., Apostol). Primes 3 and 5 have only 1 solution and no lifting to powers >= 2 is possible. See A347833 for these multiplicities.
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REFERENCES
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Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, pp 121, 122.
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LINKS
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PROG
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(PARI) isok(m) = {my(f=factor(m)); for (k=1, #f~, my(p=f[k, 1]); if ((p==3) || (p==5), if (f[k, 2] > 1, return (0)), if (kronecker(p, 15) != 1, return(0))); ); return (1); } \\ Michel Marcus, Oct 23 2021
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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