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A173062
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Primes of the form 2^r * 13^s - 1.
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4
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3, 7, 31, 103, 127, 337, 1663, 5407, 8191, 131071, 346111, 524287, 2970343, 3655807, 22151167, 109051903, 617831551, 1631461441, 2007952543, 2147483647, 32127240703, 194664464383, 275716983697, 958348132351, 1357375919743, 1670616516607, 49834102882303, 57349132609183
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OFFSET
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1,1
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COMMENTS
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s = 0 is "trivial" case of Mersenne primes: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
Mersenne prime exponents r: 2, 3, 5, 7, 13, 17, 19, 31, ...
Necessarily r odd as for r = 2*k and p a prime of form 6*n+1: 2^(2*k) * p^j - 1 a multiple of 3.
Proof by induction with 2^2 * p^1 - 1 = 4*(6*n+1) - 1 = 3*(8*n+1), 2^2(k+1) * p^j - 1 = 4* (2^k * p^j - 1) + 3.
No prime in case i = j = k (k>1) as a^k - 1 has divisor a - 1.
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REFERENCES
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Peter Bundschuh: Einfuehrung in die Zahlentheorie, Springer-Verlag GmbH Berlin, 2002)
Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications, 2005
Paulo Ribenboim, Wilfrid Keller, Joerg Richstein: Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006
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LINKS
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EXAMPLE
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2^2*13^0 - 1 = 3 = prime(2) => a(1).
2^3*13^1 - 1 = 103 = prime(27) => a(4).
2^7*13^9 - 1 = 1357375919743 = prime(50467169414) => a(25).
list of (r,s) pairs: (2,0), (3,0), (5,0), (3,1), (7,0), (1,2), (7,1), (5,2), (13,0), (17,0), (11,2), (19,0), (3,5), (7,4), (17,2), (23,1), (7,6), (1,8), (5,7), (31,0), (9,7), (19,5), (1,10), (25,4), (7,9), (11,8), (27,5), (5,11), (25,6), (19,8), (13,10), (3,13), (29,7), (5,14), (39,5), (15,13), (5,16), ...
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PROG
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(PARI) lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/13; for(s=0, logint(nn\q, 13), p=13*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ Jinyuan Wang, Feb 23 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 09 2010
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EXTENSIONS
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STATUS
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approved
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