## How many Fermat primes are there?

## How many Fermat primes are there?

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The only known Fermat primes are the first five Fermat numbers: F0=3, F1=5, F2=17, F3=257, and F4=65537. A simple heuristic shows that it is likely that these are the only Fermat primes (though many folks like Eisenstein thought otherwise).

**Is 7 a Fermat prime?**

No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime. With the exception of F0 and F1, the last digit of a Fermat number is 7.

**How do you prove Fermat numbers are relatively prime?**

Any two distinct Fermat numbers Φm and Φn with m>n are relatively prime. Proof. Let Φm and Φn be distinct Fermat numbers with m > n, and suppose that d > 0 is a common divisor of Φm and Φn, then d divides 2 = Φm − Φ0 · Φ1 ··· Φn ··· Φm−1. Therefore, d = 1 or d = 2, but Φm and Φn are odd, so we must have d = 1.

### How do you know what a prime number is?

To prove whether a number is a prime number, first try dividing it by 2, and see if you get a whole number. If you do, it can’t be a prime number. If you don’t get a whole number, next try dividing it by prime numbers: 3, 5, 7, 11 (9 is divisible by 3) and so on, always dividing by a prime number (see table below).

**Are Fermat numbers square free?**

It has been conjectured that the Fermat and Mersenne numbers are all square-free. In this note it is shown that if some Fermat or Mersenne number fails to be square-free, then for any prime p whose square divides the appropriate number, it must be that 2P”1 = 1 (mod p2).

**Is 4294967297 a prime number?**

These numbers are now called Fermat numbers, and, indeed, for small values of n, they give us 3, 5, 17, 257 and 65537, all of which are prime numbers. The next Fermat number, taking n = 5, is 4,294,967,297.

## Is Mersenne Prime 31?

, 3, 5, 7, 13, 17, 19, 31, 61, 89, (OEIS A000043). Mersenne primes were first studied because of the remarkable properties that every Mersenne prime corresponds to exactly one perfect number.

**Are all Fermat numbers coprime?**

Justification: Fermat numbers are increasing, and any two Fermat numbers are coprime. So there are at least n prime numbers less than Fn. But, Fn’s factorisation should also contain a new prime, and, obviously, that prime would be less than or equal to Fn.

**Is 31 a Mersenne prime?**

### Is 8128 a perfect number?

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Diviors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064. Therefore, 8128 is a perfect number.