Prime Possibilities

Number Theory Level 4

Find the sum of the first three (positive) prime values of $\alpha$ such that $2 \alpha ^2 -1$ is also prime and $(2 \alpha ^2 -\alpha )^2 \equiv 1 \pmod 7$ .

The answer is 22.

1 solution

Chew-Seong Cheong
Feb 19, 2015

Using the following spreadsheet,

we find that the first three primes that satisfy the two conditions are $2$ , $3$ and $17$ , therefore, the required answer is $2+3+17=\boxed{22}$ .

It's sad for a problem to be only solvable with a calculator or a computer, i didn't want to try long because, in my point of view, it had no interest.

Simon Boutin - 5 years, 11 months ago

I liked your spreadsheet, Chew-Seong! Exceptional analysis!

Guilherme Dela Corte - 6 years, 3 months ago

how to create these spreadsheets?

Dev Sharma - 5 years, 6 months ago

If you solve x^2=1 (mod 7) you can (with some work) narrow down alpha=1, 2, or 3 (mod 7). Then you can check that subset of primes (2,3,17,23,etc.) for the corresponding value of 2*alpha^2-1 to be prime to ensure all criteria are met. Doing so will find the answer Chew-Seong Cheong found using the spreadsheet, albeit with a little more labor. Sorry for the lack of a proper solution using Latex ;p

Jonathan Hocker - 4 years, 7 months ago

Prime Possibilities

The answer is 22.

1 solution

0 pending reports