Which of the following equations have a solution?
(a) ( )
(b) ( )
(c) ( )
(d) ( )
Could anyone show me how to solve these step by step? I can't just submit my answers as 'has a solution', 'has no solution.' Thanks!
These are problems based on quadratic residues.I hope you are familiar with the legendre symbol. For reference, if p is an odd prime...
(a) Compute
(b) Since (31,3) = 1, compute
(c) ( ) is same as ( ). Again we have (2,41) = 1. Therefore compute
(d)This one is interesting. You have to transform this quadratic congruence to linear by completing the squares and then find the quadratic residue
There are probably theorems involved there that I don't know, but here goes:
a) You can get this simply by a listing. No number a modulo 29 has 17 as its square. There is no solution.
b)
c)
and this has no solution.
d)
Complete the square:
(Just as a note:
So add
to both sides.)
and this has no solution for x + 13, so there is no solution for x.
-Dan
You can kill these problems with the Golden theorem.
.
First so we get . That is definitely solvable.(b) ( )
Note so .(c) ( )
.
thus . Add to both sides to get .(d) ( )