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Search: id:A089082
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| A089082 |
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Smallest prime factor of 3^n - 2^n. |
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1
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| 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
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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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 primes 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). If fact these two theorems may be generalized for a&b, (a,b) and (a^n +/- b^n)/(a +/1 b).
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EXAMPLE
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3^9 - 2^9 = 19*1009.
2^17 - 2^17 = 129009091.
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MATHEMATICA
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PrimeFactors[n_] := Flatten[ Table[ #[[1]], {1}] & /@ FactorInteger[n]]; Do[ Print[ PrimeFactors[3^n - 2^n][[1]]], {n, 2, 60}] (from Robert G. Wilson v Dec 05 2003)
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PROGRAM
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(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", ") ) }
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CROSSREFS
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Sequence in context: A091059 A139243 A139237 this_sequence A106229 A129734 A109325
Adjacent sequences: A089079 A089080 A089081 this_sequence A089083 A089084 A089085
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Dec 04 2003
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