Browse Groups

• Hi, Is it entirely obvious that 2^(p-1)+3^(p-2) -1 is always divisible by 2 and 3 where p is prime??? Bill
Message 1 of 3 , Jan 16, 2007
View Source
Hi,

Is it entirely obvious that 2^(p-1)+3^(p-2) -1 is always divisible by 2
and 3 where p is prime???

Bill
• ... 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
View Source
--- 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

() ASCII ribbon campaign () Hopeless ribbon campaign
/\ against HTML mail /\ against gratuitous bloodshed

[stolen with permission from Daniel B. Cristofani]

____________________________________________________________________________________
Cheap talk?
Check out Yahoo! Messenger's low PC-to-Phone call rates.
http://voice.yahoo.com
• ... 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 1 of 3 , Jan 16, 2007
View Source
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
Your message has been successfully submitted and would be delivered to recipients shortly.
• Changes have not been saved
Press OK to abandon changes or Cancel to continue editing
• Your browser is not supported
Kindly note that Groups does not support 7.0 or earlier versions of Internet Explorer. We recommend upgrading to the latest Internet Explorer, Google Chrome, or Firefox. If you are using IE 9 or later, make sure you turn off Compatibility View.