I was working on he Collatz Conjecture and noticed that the reverse of 3x+1 is x-1/3 This is possible for Powers of 2 that end in 6 and 4. Where the result will have in the case of 6 a factor of 5 and in the case of 4 no factor of five. From observation naturally. My question is do all Mersenne Primes come from powers of 2 that end in 2 or 8 ? I am at a loss for any search results on this.
Asked
Active
Viewed 48 times
2
-
Naturally. I am limited on the print out aspect so I thought I'd ask. – Ernst D. Berg Jan 06 '18 at 03:04
1 Answers
4
No, it is almost true. The powers of $2 \bmod 10$ cycle with period $\phi(10)=4$, where $\phi(n)$ is Euler's totient function. Mersenne primes are of the form $2^n-1$, where $n$ is prime. As almost all primes are either $1 \bmod 4$ or $3 \bmod 4$, the last digit of the power of $2$ is equivalent to $2^1=2 \bmod 10$ or $2^3=8 \bmod 10$. The one exception is $2^2-1=3$, which comes from a power of $2$ that does not end in $2$ or $8$.
Ross Millikan
- 374,822