A353171 - OEIS

A353171

Irregular triangle read by rows; T(n,k) = 2^k (mod prime(n)), terminating when T(n,k) = 1.

-1, 1, 2, -1, -2, 1, 2, -3, 1, 2, 4, -3, 5, -1, -2, -4, 3, -5, 1, 2, 4, -5, 3, 6, -1, -2, -4, 5, -3, -6, 1, 2, 4, 8, -1, -2, -4, -8, 1, 2, 4, 8, -3, -6, 7, -5, 9, -1, -2, -4, -8, 3, 6, -7, 5, -9, 1, 2, 4, 8, -7, 9, -5, -10, 3, 6, -11, 1, 2, 4, 8, -13, 3, 6, 12, -5, -10, 9, -11, 7, 14, -1, -2, -4, -8, 13, -3, -6, -12, 5, 10, -9, 11, -7, -14, 1, 2, 4, 8, -15, 1

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OFFSET

2,3

COMMENTS

Although the most significant digits of powers of 2 in base n are generally not periodic (the exception being when n is a power of 2), the least significant digits are. For example, 2 to an even power is congruent to 1 (mod 3) and 2 to an odd power is congruent to -1 (mod 3). This means that one can determine one of the prime factors of a Mersenne number, A000225, using the exponent. If n == 0 (mod 2), then A000225(n) == 0 (mod 3) (is a multiple of 3); if n == 0 (mod 4), then A000225(n) == 0 (mod 5); if n == 0 (mod 3), then A000225(n) == 0 (mod 7), and so on.

This general fact gives a reason for why certain Mersenne numbers are not prime (even with prime exponents). If p is congruent to 0 mod A014664(n) (the length of an n-th row) and prime(n) is less than the A000225(p), then prime(n) is a nontrivial factor of A000225(p).

LINKS

Table of n, a(n) for n=2..102.

EXAMPLE

Irregular triangle begins

n/k|| 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ... || Length ||

----------------------------------------------------------------------

2 || -1 1 || 2 ||

3 || 2, -1, -2, 1 || 4 ||

4 || 2, -3, 1 || 3 ||

5 || 2, 4, -3, 5, -1, -2, -4, 3, -5, 1 || 10 ||

6 || 2, 4, -5, 3, 6, -1, -2, -4, 5, -3, -6, 1 || 12 ||

7 || 2, 4, 8, -1, -2, -4, -8, 1 || 8 ||