A307436 a(n) is the smallest k such that (Z/kZ)* contains C_(2n) X C_(2n) as a subgroup, where (Z/kZ)* is the multiplicative group of integers modulo n.

8, 65, 63, 544, 275, 455, 1247, 1088, 513, 1025, 1541, 7081, 4187, 3277, 1891, 12416, 14111, 2701, 43739, 7667, 2107, 10235, 6533, 11543, 12625, 8321, 8829, 31753, 13747, 8723, 116003, 49408, 10787, 39593, 14981, 23579, 33227, 104653, 12403, 45067, 61337

Offset: 1

Author

Jianing Song, Apr 08 2019

Keywords

• nonn

Comments

a(n) exists for all n: by Dirichlet's theorem on arithmetic progressions, there exists two primes p, q congruent to 1 modulo 2n, in which case C_(2n) X C_(2n) is a subgroup of (Z/(p*q)Z)*.

a(n) is the smallest k such that there exists some x, y such that ord(x,k) = ord(y,k) = 2n and the set of powers of x and the set of powers of y modulo k have trivial intersection {1}, where ord(x,k) is the multiplicative order of x modulo n.

Examples

			(Z/8Z)* = C_2 X C_2, in which {3^e mod 8} = {3, 1} and {3^e mod 8} = {5, 1};
(Z/65Z)* = C_4 X C_12, in which {8^e mod 65} = {8, 64, 57, 1} and {12^e mod 65} = {12, 14, 38, 1};
(Z/63Z)* = C_6 X C_6, in which {2^e mod 63} = {2. 4. 8, 16, 32, 1} and {5^e mod 63} = {5, 25, 62, 58, 38, 1};
(Z/544Z)* = C_2 X C_8 X C_16, in which {9^e mod 544} = {9, 81, 185, 33, 297, 497, 121, 1} and {13^e mod 544} = {13, 169, 21, 273, 285, 441, 293, 1};
(Z/275Z)* = C_10 X C_20, in which {4^e mod 275} = {4, 16, 64, 256, 199, 246, 159, 86, 69, 1} and {6^e mod 275} = {6, 36, 216, 196, 76, 181, 261, 191, 46, 1};
(Z/455Z)* = C_2 X C_12 X C_12, in which {2^e mod 455} = {2, 4, 8, 16, 32, 64, 128, 256, 57, 114, 228, 1} and {3^e mod 455} = {3, 9, 27, 81, 243, 274, 367, 191, 118, 354, 152, 1}.
		

Links

• Jianing Song, Table of n, a(n) for n = 1..200
• Wikipedia, Multiplicative group of integers modulo n

Crossrefs

Cf. A307437.

Programs

PARI

a(n) = my(d=1, v=znstar(d)[2]); while(sum(i=1, #v, !(v[i]%(2*n)))<2, d++; v=znstar(d)[2]); d