1) Show that a prime divisorpof the Fermat number must be of the form .

(Hint: Show that . Then show that is congruent to 1 (mod p). Conclude that (p-1)/2)

Printable View

- November 10th 2010, 05:19 PMJanu42Quadratic Residue Question
1) Show that a prime divisor

*p*of the Fermat number must be of the form .

(Hint: Show that . Then show that is congruent to 1 (mod p). Conclude that (p-1)/2) - November 11th 2010, 10:26 AMBruno J.
- November 11th 2010, 12:29 PMJanu42
- November 11th 2010, 04:50 PMBruno J.
Well, if has order modulo , i.e. is the least positive integer such that , then for any exponent such that , we must have .

Hence if , this means that the order of must divide ...

Extra hint : to show that , you can use the "second supplement" to the quadratic reciprocity law, which states that for any odd prime .