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• ... Humans and mail are slow. Computers and Pari/GP are fast. ? for(i=2,12,print(i : 2^(i-1)%6 3^(i-2)%6 (2^(i-1)+3^(i-2))%6)) 2 : 2 1 3 3 : 4 3 1 4 : 2
Message 1 of 3 , Jan 16, 2007
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--- leavemsg1 <leavemsg1@...> wrote:
> Is it entirely obvious that 2^(p-1)+3^(p-2) -1 is always divisible by 2
> and 3 where p is prime???

Humans and mail are slow. Computers and Pari/GP are fast.

? for(i=2,12,print(i" : "2^(i-1)%6" "3^(i-2)%6" "(2^(i-1)+3^(i-2))%6))
2 : 2 1 3
3 : 4 3 1
4 : 2 3 5
5 : 4 3 1
6 : 2 3 5
7 : 4 3 1
8 : 2 3 5
9 : 4 3 1
10 : 2 3 5
11 : 4 3 1
12 : 2 3 5

So it's not just true when p's an (odd) prime, but true whenever p's any odd
number.

Phil

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• ... Yes. 2^whatever is even, 3^whatever is odd, so 2^x + 3^y - 1 is even. 2^even = 1 (mod 3), 3^whatever = 0 (mod 3), so 2^even + 3^whatever - 1 is divisible
Message 2 of 3 , Jan 16, 2007
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On 1/16/07, leavemsg1 <leavemsg1@...> wrote:
> Is it entirely obvious that 2^(p-1)+3^(p-2) -1 is always divisible by 2
> and 3 where p is prime???

Yes.

2^whatever is even, 3^whatever is odd, so 2^x + 3^y - 1 is even.

2^even = 1 (mod 3), 3^whatever = 0 (mod 3), so
2^even + 3^whatever - 1 is divisible by 3.

So if p is odd (not necessarily prime, and not 2), then the
divisibility will happen as you have said.

--Joshua ucker
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