A089082 Smallest prime factor of 3^n - 2^n.

5, 19, 5, 211, 5, 29, 5, 19, 5, 23, 5, 53, 5, 19, 5, 129009091, 5, 1559, 5, 19, 5, 47, 5, 101, 5, 19, 5, 68629840493971, 5, 617671248800299, 5, 19, 5, 29, 5, 8891471, 5, 19, 5, 821, 5, 431, 5, 19, 5, 1129, 5, 29, 5, 19, 5, 19383245658672820642055731, 5, 23, 5, 19, 5

Offset

2,1

Comments

Theorem I. If n is prime and the prime factors of b^n - (b-1)^n are p_1, p_2, p_3, ..., p_i & i <=1, then p_i == 1 (mod n) for all i's. This is a result from Fermat's little theorem.

Theorem II. If n is not prime and n = d_1 * d_2 * d_3 * ... * d_j and the prime factors of b^n - (b-1)^n are p_1, p_2, p_3, ..., p_i & i < 1, then some prime factors, q_1, q_2, q_3, ..., q_k & k <= 1 are not primitive, i.e., they are prime factors of b^d_j - (b-1)^d_j. Excluding from the list of p_i those which are not primitive, then the rest are also == 1 (mod n). In fact, these two theorems may be generalized for a and b, (a,b) and (a^n +- b^n)/(a +- b).

Links

Chai Wah Wu, Table of n, a(n) for n = 2..598

Example

3^9 - 2^9 = 19*1009.
3^17 - 2^17 = 129009091.

Mathematica

PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; Do[ Print[ PrimeFactors[3^n - 2^n][[1]]], {n, 2, 60}] (* Robert G. Wilson v, Dec 05 2003 *)

Prog

(PARI) leastfactor(a, n) = { for(x=2, n, y = a^x-(a-1)^x; f = factor(y); v = component(component(f, 1), 1); print1(v", ") ) }

Crossrefs

Sequence in context: A318181 A274995 A370491 * A217011 A106229 A129734

Adjacent sequences: A089079 A089080 A089081 * A089083 A089084 A089085

Keyword

nonn

Author

Cino Hilliard, Dec 04 2003

Status

approved