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

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Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (33,759,348)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (44,987,733)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (457,900,406)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (2,072,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 153541 - Posted: 24 Jan 2022 | 22:53:04 UTC
Last modified: 24 Jan 2022 | 23:13:27 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.

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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (15,959,634)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,228,147)WW Turquoise: Earned 5,000,000 credits (9,640,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
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

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Joined: 5 Apr 14
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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (593,283)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (174,818)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (112,799)PPS LLR Jade: Earned 10,000,000 credits (15,959,634)PSP LLR Silver: Earned 100,000 credits (428,457)SoB LLR Silver: Earned 100,000 credits (466,812)SR5 LLR Silver: Earned 100,000 credits (210,142)SGS LLR Silver: Earned 100,000 credits (112,277)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve (suspended) Silver: Earned 100,000 credits (175,037)PPS Sieve Bronze: Earned 10,000 credits (10,113)AP 26/27 Bronze: Earned 10,000 credits (12,129)GFN Ruby: Earned 2,000,000 credits (4,228,147)WW Turquoise: Earned 5,000,000 credits (9,640,000)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 153543 - Posted: 24 Jan 2022 | 23:50:58 UTC

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

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Joined: 16 Feb 10
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ID: 55391
Credit: 879,366,379
RAC: 316,449
Discovered 2 mega primesFound 1 prime in the 2018 Tour de PrimesFound 1 prime in the 2022 Tour de Primes321 LLR Turquoise: Earned 5,000,000 credits (6,055,323)Cullen LLR Gold: Earned 500,000 credits (776,297)ESP LLR Ruby: Earned 2,000,000 credits (3,433,680)Generalized Cullen/Woodall LLR Ruby: Earned 2,000,000 credits (2,443,837)PPS LLR Sapphire: Earned 20,000,000 credits (33,759,348)PSP LLR Turquoise: Earned 5,000,000 credits (6,587,988)SoB LLR Sapphire: Earned 20,000,000 credits (44,987,733)SR5 LLR Turquoise: Earned 5,000,000 credits (6,205,694)SGS LLR Ruby: Earned 2,000,000 credits (3,627,819)TRP LLR Turquoise: Earned 5,000,000 credits (7,078,152)Woodall LLR Amethyst: Earned 1,000,000 credits (1,693,614)321 Sieve (suspended) Emerald: Earned 50,000,000 credits (50,256,050)Cullen/Woodall Sieve (suspended) Turquoise: Earned 5,000,000 credits (5,571,178)Generalized Cullen/Woodall Sieve (suspended) Emerald: Earned 50,000,000 credits (50,009,610)PPS Sieve Double Silver: Earned 200,000,000 credits (457,900,406)Sierpinski (ESP/PSP/SoB) Sieve (suspended) Jade: Earned 10,000,000 credits (10,165,888)TRP Sieve (suspended) Sapphire: Earned 20,000,000 credits (20,071,454)AP 26/27 Turquoise: Earned 5,000,000 credits (6,798,063)GFN Emerald: Earned 50,000,000 credits (57,113,430)WW Ruby: Earned 2,000,000 credits (2,072,000)PSA Double Bronze: Earned 100,000,000 credits (102,762,384)
Message 153544 - Posted: 25 Jan 2022 | 0:02:11 UTC
Last modified: 25 Jan 2022 | 0:02:31 UTC

Ok, thanks. That makes perfect sense.
I noticed the recurrence relation for Fn but didn't make the obvious connection to the Mersenne Numbers: Fn = M2n + 2

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

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