Problems & Puzzles: Puzzles

 Puzzle 1121 Phi(p)/Phi(p-2)=2   Sebastián Martín Ruiz sent the following nice puzzle:   I have obtained the following results for the Fermat Numbers, p=2^(2^n)+1,  being Phi the Euler Totient Function: n= 0 Phi[p]/Phi[p-2]= 2 n= 1 Phi[p]/Phi[p-2]= 2 n= 2 Phi[p]/Phi[p-2]= 2 n= 3 Phi[p]/Phi[p-2]= 2 n= 4 Phi[p]/Phi[p-2]= 2 n= 5 Phi[p]/Phi[p-2]= 261735/131072 =1.99688... n= 6 Phi[p]/Phi[p-2]= 18764930006193/9367860543488 = 2.00312... n= 7 Phi[p]/Phi[p-2]= 9615364552842273224968401793233/4800163769175502470780823797760 = 2.00313... It is observed that in the case where p is prime, that is, from n=0 to 4 Phi(p)/Phi(p-2)=2 and if p is composite, that is, from n=5 to 7 Phi(p)/Phi(p-2) is not an exact number close to 2. Q) Let p a Fermat Number. Prove p is a Fermat prime if and only if Phi(p)/Phi(p-2)=2, or find a counterexample.

During the week 4-10 Feb 2023, contributions came from Emmanuel Vantieghem

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Emmanuel wrote:

The conjecture is not true.  This is my proof :

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