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In 1995, Yannick Saouter produced the study of a family of numbers close to the Fermat numbers: [tex]A_n=4^{3^n}+2^{3^n}+1[/tex] .

(See: http://www.inria.fr/rrrt/rr-2728.html)

Saouter proved that this A_n serie shares many properties with the Fermat numbers:

3.4 A_n numbers are pairwise relatively primes

3.3 [tex]A_n \text{ is prime iff } 5^{(A_n-1)/2} \equiv -1 \pmod{A_n}[/tex]

3.5 [tex]p | A_n ==> p = 1 \pmod{2.3^{n+1}}[/tex]

I've also discovered (and checked with PARI) that the following property is true:

[tex]A_n = 3+2(2^{3^{\scriptstyle n}-1}+1)\prod_{i=0}^{n-1}A_i[/tex]

(I've summarized the properties at: http://tony.reix.free.fr/Mersenne/PropertiesOfFermatLikeTNumbers.pdf)

A VERY interesting thing is that the primality of these numbers can be checked with the Pépin's test, with 5 instead of 3, like Fermat numbers.

Saouter provides the divisors of several of these numbers (n up to 39).

It appears:

- that 10 of them have no divisors known

- only the 3 first Saouter numbes are prime (same as Fermat numbers).

Is someone interested in studying these numbers in more details ?

Is it possible to adapt some Pépin's test code to this kind of numbers ?

Regards,

Tony

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# A VERY interesting Fermat-like sequence: A_n=4^3^n+2^3^n+1

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