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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