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A319099
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Number of solutions to x^5 == 1 (mod n).
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10
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1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1
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OFFSET
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1,11
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COMMENTS
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All terms are powers of 5. Those n such that a(n) > 1 are in A066500.
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LINKS
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FORMULA
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Multiplicative with a(5) = 1, a(5^e) = 5 if e >= 2; for other primes p, a(p^e) = 5 if p == 1 (mod 5), a(p^e) = 1 otherwise.
If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(5, k_i).
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EXAMPLE
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Solutions to x^5 == 1 (mod 11): x == 1, 3, 4, 5, 9 (mod 11).
Solutions to x^5 == 1 (mod 25): x == 1, 6, 11, 16, 21 (mod 25) (x == 1 (mod 5)).
Solutions to x^5 == 1 (mod 31): x == 1, 2, 4, 8, 16 (mod 31).
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MATHEMATICA
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f[p_, e_] := If[Mod[p, 5] == 1, 5, 1]; f[5, 1] = 1; f[5, e_] := 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)
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PROG
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(PARI) a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(5, Z[i]));
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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