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A173321
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a(n) = 4*n! - 1.
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7
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3, 3, 7, 23, 95, 479, 2879, 20159, 161279, 1451519, 14515199, 159667199, 1916006399, 24908083199, 348713164799, 5230697471999, 83691159551999, 1422749712383999, 25609494822911999, 486580401635327999
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OFFSET
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0,1
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COMMENTS
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With this sequence, it is possible to prove that there are infinitely many prime numbers of the form 4*k+3.
Prove that:
1. Every prime factor of a(n) is > n, and,
2. All these prime factors are of the form 4*k+1 or 4*k+3.
3. There is at least one prime of the form 4*k+3 > n,
4. The set of prime numbers of the form 4*k+3 is infinite.
(End)
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REFERENCES
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Transmath, Term S, Spécialité, Programme 2002, Nathan, 2002, Exercice 82 p. 93.
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LINKS
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FORMULA
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MAPLE
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MATHEMATICA
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PROG
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(Magma) [4*Factorial(n)-1: n in [0..25]]; /* or */ [3] cat [n eq 1 select n+2 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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