P153: A Prime Proof

  Prove that whenever P and P*P+2 are both primes,
  P*P+4 is also prime. (P+2 is prime too (Torsten).)


Variation I: (Torsten Sillke)

  Prove that whenever P and P^4-6 are both primes,
  P^4+6 is also prime.


Variation II: (Torsten Sillke)

  Prove that whenever P, P^3-6, and P^3+6 are all primes,
  P^2-2 is also prime.



A puzzle from:

Angela Fox Dunn,
  Second Book of Mathematical Bafflers,
  Dover Publ., 1983

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Hint Allert













































Hint:
  A Prime Proof: The number 3 is the key.
  Variation I: The number 5 is the key.
  Variation II: The number 7 is the key.






















































Generalization:
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(A) Let q be a prime then for each integer n we have:

      q | n   or   q | n^(q-1) + (q-1)

    This is the little theorem of Fermat.

    Therefore the set

      { n | n  and  n^(q-1) + (q-1)  are primes } is a subset of {q}.

    
    Application:

    q=3:   { n | n  and  n*n + 2  are primes } = {3}
    q=5:   { n | n  and  n^4 + 4  are primes } = {}



(B) Let q be an odd prime then for each integer n we have:

      q | n   or   q | n^((q-1)/2) + (q-1)   or   q | n^((q-1)/2) - (q-1)

    Application:

    q=7:   { n | n  and  n^3 + 6  and  n^3 - 6  are primes } = {7}

Reference:

Hon73: Ross Honsberger;
   Mathematical Gems I,
   The Dolciani Mathematical Expositions No. 1
   The Mathematical Association of America, 1973
   (german: Mathematische Edelsteine, Vieweg, 1981)
   - Chapter 1: an old chinease theorem and Pierre de Fermat

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