00:01
We're asked to show that if 2 to the m plus 1 is an odd prime, then m is equal to 2 to the m for some non -negative integer n.
00:16
So first, let's suppose that 2 to the m plus 1 is an odd prime.
00:33
Now 2 is congruent to negative 1 mod 3.
00:44
So it follows that 2 to the m is congruent to negative 1 to the m odd 3.
00:57
And therefore it follows that 2 to the m plus 1 our number is congruent to negative 1 to the m plus 1 mod 3.
01:23
So we have that 2 to the m plus 1 is congruent to 2 mod 3 when m is even and we have that 2 to the m plus 1 is congruent to 0, mod 3, when m is odd, since negative 1 v .m is negative 1.
02:12
Now, 2 to the m plus 1 congruent to 0 is impossible.
02:32
For if it were the case, then it would have followed that 3 would divide 2 of the m plus 1.
02:50
But recall that we have that 2 to the m plus 1 is an odd prime, and so we have to be that 2 to the m plus 1 would have to equal three, or that 2 to the m would have to equal 2.
03:40
I wouldn't say this is actually impossible, but if this is the case, if 2 of the m plus 1 is congruent to 0 mod 3, then it follows that 3 divides 2 to the m plus 1.
03:57
So 2 to the m plus 1 has to be equal to 3 since 2 to the m plus 1 is prime, and therefore 2 to the m equals 2 so that m is equal to 1.
04:10
So it follows that in this case 2 to the m plus 1 is prime and m is equal to 1 which is the same as 2 to the 0.
04:24
And 0 is a non -negative integer.
04:45
Now let's consider the other case where m is even.
04:54
So if 2 to the m plus 1 is congruent to 2, mod 3, what does this imply? now since m is even, you can find an integer case, such that m equals 2k.
05:34
So now let's prove this by breaking this up into cases.
05:42
So suppose that k is equal to 1, in fact these are positive integers k, then we have that m is equal to 2, and we have that 2 squared plus 1 is 5, which is equal to 2 plus 3 times 1 which is clearly congruent to 2 mod 3.
06:19
So it's true in this case.
06:21
Now let's suppose that k is odd.
06:30
Then 2 to the m plus 1 is equal to 2 to the 2 k plus 1 and because k is odd it follows that 2 times k is also odd and we can factor this at as 2 squared plus 1 times 2 to the k, sorry 2 to the 2 raised to the m minus 1 power.
08:03
This is because k is odd, minus 2 to the 2 raised to the m 2 minus 1 power.
08:29
And then we'll be alternating signs...