A281728 Johnson's non-Wieferich numbers of the first kind.

3, 11, 13, 43, 241, 683, 2731, 43691, 61681, 174763, 2796203, 15790321, 715827883, 4278255361, 2932031007403, 4363953127297

Offset: 1

Author

Felix Fröhlich, Jan 28 2017

Keywords

• nonn
• more

Comments

This is the case a = 2 of primes p such that p-1 has the a-adic expansion bb...b00...0bb...b00...0_a, where b = a-1 with each of the t blocks of digits b or 0 having length k and additionally q_a == (a^k + 1)/(t + 1)*k =/= 0 (mod p), where q_a denotes the Fermat quotient to base a (cf. Johnson, 1977).

These are prime numbers of the form (2^m + 1)/(2^n + 1). Note that if m,n > 0, then 2^n + 1 divides 2^m + 1 if and only if m/n is odd. - Thomas Ordowski, Feb 17 2024

Examples

			(2^49+1)/(2^7+1) = 4363953127297 = 111111100000001111111000000011111110000001.
		

Links

• J. B. Dobson, A note on the two known Wieferich Primes (see section "Johnson's non-Wieferich numbers of the first kind (his Corollary 5 with a = 2, b = 1)").
• Wells Johnson, On the nonvanishing of Fermat quotients (mod p), Journal f. die reine und angewandte Mathematik 292, (1977): 196-200.

Crossrefs

Cf. A370425 (integers of the form (2^m+1)/(2^n+1)).