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Message boards : General discussion : Why does p divide 2^(p-1)-1?

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Profile BurProject donor
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Joined: 25 Feb 20
Posts: 332
ID: 1241833
Credit: 22,611,276
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321 LLR Gold: Earned 500,000 credits (538,216)Cullen LLR Amethyst: Earned 1,000,000 credits (1,169,946)ESP LLR Gold: Earned 500,000 credits (636,842)Generalized Cullen/Woodall LLR Silver: Earned 100,000 credits (212,232)PPS LLR Gold: Earned 500,000 credits (883,715)PSP LLR Gold: Earned 500,000 credits (663,928)SoB LLR Silver: Earned 100,000 credits (217,346)SR5 LLR Gold: Earned 500,000 credits (531,229)SGS LLR Amethyst: Earned 1,000,000 credits (1,042,382)TRP LLR Gold: Earned 500,000 credits (561,429)Woodall LLR Gold: Earned 500,000 credits (781,741)321 Sieve Ruby: Earned 2,000,000 credits (2,107,153)PPS Sieve Amethyst: Earned 1,000,000 credits (1,045,010)AP 26/27 Ruby: Earned 2,000,000 credits (2,470,273)GFN Turquoise: Earned 5,000,000 credits (7,129,018)PSA Silver: Earned 100,000 credits (244,815)
Message 147019 - Posted: 24 Dec 2020 | 7:17:02 UTC

This might sound like a strange question and I am not looking for a mathmatical proof. But maybe there is some more accessible way to show why there is a relation at all? It seems so arbitrary to me, which, of course, it actually isn't.
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Primes: 1281979 & 12+8+1979 & 1+2+8+1+9+7+9 & 1^2+2^2+8^2+1^2+9^2+7^2+9^2 & 12*8+19*79 & 12^8-1979 & 1281979 + 4 (cousin prime)

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Profile JeppeSNProject donor
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Found 1 prime in the 2020 Tour de Primes321 LLR Gold: Earned 500,000 credits (529,293)Cullen LLR Gold: Earned 500,000 credits (611,298)ESP LLR Silver: Earned 100,000 credits (139,922)Generalized Cullen/Woodall LLR Bronze: Earned 10,000 credits (35,236)PPS LLR Turquoise: Earned 5,000,000 credits (9,594,179)PSP LLR Silver: Earned 100,000 credits (212,242)SoB LLR Silver: Earned 100,000 credits (237,390)SR5 LLR Silver: Earned 100,000 credits (145,419)SGS LLR Silver: Earned 100,000 credits (105,212)TRP LLR Silver: Earned 100,000 credits (342,501)Woodall LLR Silver: Earned 100,000 credits (109,455)321 Sieve 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 Amethyst: Earned 1,000,000 credits (1,674,106)PSA Turquoise: Earned 5,000,000 credits (7,614,290)
Message 147024 - Posted: 24 Dec 2020 | 10:25:03 UTC - in response to Message 147019.

This might sound like a strange question and I am not looking for a mathmatical proof. But maybe there is some more accessible way to show why there is a relation at all? It seems so arbitrary to me, which, of course, it actually isn't.

Not sure what kind of answer you are after, when not a proof.

This result is known as Fermat's little theorem (Wikipedia). I suggest you read about it. What they call a, is 2 in your example, so the result is valid if 2 does not divide the prime p.

A modern way of understanding the result, is that the (nonzero) integers modulo p form a group with multiplication, and in that group, 2 is a member; and the order of a member divides the order of the group which is p-1 (Lagrange's theorem).

/JeppeSN

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Message boards : General discussion : Why does p divide 2^(p-1)-1?

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