Hi
I need direction with the following problem.
p<>2 is a prime number, suppose the equation x^2=-1(modp) has a solution.
Show that p=1(mod4).
SK
You could use complex numbers to prove this.
First note that if , then is prime. Now it will suffice to show that if has a solution, then is not prime. (This will force .
If the equation is satisfied, then there exists an integer such that . We can factor the right hand side to receive . If were prime, then or , but this would mean that , which is absurd. Therefore, is not prime.
Let be a solution of . Therefor , from Fermat's theorem:
There is impossible that , because if then:
, and therefore , and the conclusion has , which is not true. Hence is from the form of .
The end!
By the way...
It also true that:
Let is a prime number, the equation has a solution if and only if
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