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A093428
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Prime numbers which are successors of a power of a Fibonacci number.
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0
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5, 17, 257, 65537, 1336337, 19170731299728100000001, 285347346718226949041792907369577650673693754163660005676181161059099319730177, 29585383599687066848440635756425168157198788892517565295922752892368299949134315777
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Let a(n) = Fibonacci(x)^y+1, then there exists some a,b > 0, such that x = 3*a and y = 2^b. For the example a(5) = 1336337: x = 9, y = 4, a = 3 and b = 2.
Last digit seems to be usually 7, except for a(1) and a(6). - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 09 2006
a(6) = Fibonacci(15)^8 + 1, a(7) = Fibonacci(48)^8 + 1, a(8) = Fibonacci(51)^8 + 1, a(9) = Fibonacci(63)^8 + 1, a(10) = Fibonacci(21)^32 + 1, a(11) = Fibonacci(198)^4 + 1, a(12) = Fibonacci(204)^8 + 1, a(13) = Fibonacci(366)^8 + 1. - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 09 2006
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REFERENCES
| Composites and Primes Among Powers of Fibonacci Numbers increased or decreased by one V E Hoggatt Jr and M Bicknell-Johnson, Fibonacci Quarterly vol. 15 (1977), page 2.
Greatest Common Divisors in Altered Fibonacci Sequences U Dudley, B Tucker Fibonacci Quarterly 1971, pages 89-91.
Letter from Toby Gee in Mathematical Spectrum, vol. 29 (1996/1997), page 68.
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LINKS
| R. Knott, The Mathematical Magic of the Fibonacci Numbers.
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EXAMPLE
| a(5) = 1336337 because 1336337 is prime,
and 1336337-1 = 1336336 = 34^4+1 = Fibonacci(9)^4+1
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MATHEMATICA
| Select[Take[Intersection[Flatten[Table[Fibonacci[3n]^(2^m)+1, {n, 1, 300}, {m, 1, 7}]]], {1, 400}], PrimeQ] - Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 09 2006
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CROSSREFS
| Cf. A005478, A001605, A000045.
Sequence in context: A191500 A089894 A077718 * A081479 A096996 A070294
Adjacent sequences: A093425 A093426 A093427 * A093429 A093430 A093431
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KEYWORD
| nonn
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AUTHOR
| Lior Manor (lior.manor(AT)gmail.com) May 12 2004
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EXTENSIONS
| More terms from Alexander Adamchuk (alex(AT)kolmogorov.com), Aug 09 2006
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