A152676 a(n) = A002144(n) - A002314(n).

3, 8, 13, 17, 31, 32, 30, 50, 46, 55, 75, 91, 76, 98, 100, 105, 129, 93, 162, 112, 183, 122, 144, 177, 241, 187, 217, 228, 155, 288, 203, 189, 213, 311, 269, 274, 334, 381, 266, 392, 254, 382, 348, 413, 301, 286, 489, 439, 483, 553, 516, 476, 578, 423, 487, 504

Offset: 1

Author

Artur Jasinski, Dec 10 2008

Keywords

• nonn

Comments

For the four numbers {1, A002314(n), A152676(n), A152680(n)}, the multiplication table modulo A002144(n) is isomorphic with the Latin square

1 2 3 4

2 4 1 3

3 1 4 2

4 3 2 1

and is isomorphic with the multiplication table for {1,i,-i,-1} where i = sqrt(-1), A152680(n) is isomorphic with -1, A002314(n) with i or -i and A152676(n) vice versa -i or i.

1, A002314(n), A152676(n), A152680(n) are a subfield of the Galois Field [A002144(n)].

Let p = A002144(n), the n-th prime of the form 4k+1. Then a(n) and A002314(n) are the two square roots of -1 (mod p). Note that a(n) is also the multiplicative inverse of A002314(n) (mod p). - T. D. Noe, Feb 18 2010

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. See Vol. 1, pp. 1-21.

Crossrefs

Cf. A002144, A002314, A152680.

Programs

Mathematica

aa = {}; Do[If[Mod[Prime[n], 4] == 1, k = 1; While[ ! Mod[k^2 + 1, Prime[n]] == 0, k++ ]; AppendTo[aa, Prime[n] - k]], {n, 1, 200}]; aa