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

• T.Rex
In summary, Saouter proved that a family of numbers close to the Fermat numbers share many properties with the Fermat numbers, including that they are prime iff 5^{(A_n-1)/2} \equiv -1 \pmod{A_n}. He provides the divisors of several of these numbers (n up to 39).
T.Rex
Hi,
In 1995, Yannick Saouter produced the study of a family of numbers close to the Fermat numbers: $$A_n=4^{3^n}+2^{3^n}+1$$ .
(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 $$A_n \text{ is prime iff } 5^{(A_n-1)/2} \equiv -1 \pmod{A_n}$$

3.5 $$p | A_n ==> p = 1 \pmod{2.3^{n+1}}$$

I've also discovered (and checked with PARI) that the following property is true:
$$A_n = 3+2(2^{3^{\scriptstyle n}-1}+1)\prod_{i=0}^{n-1}A_i$$

(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

Erm interesting, but due to their size it's a bit computationally awkward to calculate isn't it? I mean I'm sure there are quite a few sequences that behave in similar ways but of such great size that they can't be calculated in any short amount of time.

Not criticizing you at all, it's something I couldn't do, just wondering.

You're right. These numbers grow awfully fast, much faster than Fermat numbers.
What I think really interesting is that they share several properties with Fermat numbers. Do they also seem to have a finite number of primes ? What about the equivalent of Sierpinski's problem Saouter has studied ? Maybe a proof for Saouter numbers could help for Fermat numbers ?
Tony

Hi Zurtex,
I've updated the paper with some proof and with the number of digits of A_n.
For n=16, A_n has ~26 millions of digits. Very big !
Tony

## 1. What is a Fermat-like sequence?

A Fermat-like sequence is a sequence of numbers that follows a similar pattern to the famous Fermat's Last Theorem, which states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

## 2. How is A_n=4^3^n+2^3^n+1 calculated?

The sequence A_n=4^3^n+2^3^n+1 is calculated by raising the numbers 4 and 2 to the power of 3^n and then adding them together with 1. This results in a sequence of numbers with a similar pattern to Fermat's Last Theorem.

## 3. What makes this Fermat-like sequence interesting?

This sequence is interesting because it follows a similar pattern to Fermat's Last Theorem, but with different numbers. It also has some unique properties and has been studied by mathematicians to understand its behavior and potential connections to other mathematical concepts.

## 4. Are there any known applications of this Fermat-like sequence?

Currently, there are no known practical applications of this Fermat-like sequence. However, it is still a subject of ongoing research and could potentially have connections to other areas of mathematics or science in the future.

## 5. Is there a limit to how many terms can be calculated in this sequence?

Since the numbers in this sequence are exponentially increasing, there is a limit to how many terms can be calculated using the current computing power. However, theoretically, this sequence can continue infinitely as long as there are enough computing resources to calculate the terms.

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