## Other

drummers-lowrise

Message boards : General discussion : An Identity Concerning Fermat Numbers and Mersenne Numbers

Author Message
composite
Volunteer tester

Joined: 16 Feb 10
Posts: 1002
ID: 55391
Credit: 879,366,379
RAC: 316,449

Message 153541 - Posted: 24 Jan 2022 | 22:53:04 UTC

Fermat Numbers have the form Fk=22k+1
Mersenne Numbers have the form Mk=2k-1

*NB: these are not the sets of Fermat Primes and Mersenne Primes.

I observe the following relation: the product of the sequence of the first n Fermat Numbers is Mersenne Number 2n

PRODUCT(i=0..n-1) Fi = M2n

Is this a well-known identity? I'd rather not try to prove this if it is.

I verified it in a WxMaxima loop by multiplying the Fi and factoring the (product+1),
until Lisp got a value stack overflow, and of course it all checked out, or I wouldn't be posting this.
The last result produced before the overflow was n=18, which is M262144,
a number 78'914 digits long (ending in 5 as it should, since all the Fi are odd and F1 = 5).

EDIT: I saw no mention of this identity in OEIS.

JeppeSN

Joined: 5 Apr 14
Posts: 1695
ID: 306875
Credit: 40,800,650
RAC: 11,855

Message 153542 - Posted: 24 Jan 2022 | 23:42:33 UTC

Yes, and on the page The Prime Glossary: Fermat number, it is written like this:

F0 F1 F2 ... Fn-1 + 2 = Fn

As you see, this is one way of seeing that no number d>1 can divide both Fn and Fi for an i<n.

Also mentioned on Wikipedia: Fermat number ยง Basic properties.

/JeppeSN

JeppeSN

Joined: 5 Apr 14
Posts: 1695
ID: 306875
Credit: 40,800,650
RAC: 11,855

Message 153543 - Posted: 24 Jan 2022 | 23:50:58 UTC

You could also be interested in: Wikipedia: 4,294,967,295. /JeppeSN

composite
Volunteer tester

Joined: 16 Feb 10
Posts: 1002
ID: 55391
Credit: 879,366,379
RAC: 316,449

Message 153544 - Posted: 25 Jan 2022 | 0:02:11 UTC