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Let p be prime. Show that the equation x^2 is congruent to 1(mod p) has just...

Let p be prime. Show that the equation x^2 is congruent to 1(mod p) has just two solutions in Zp (the set of integers). We cannot use groups.

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Answer #1

Ok as you said not to use concept of group here.

Given :    

To prove: It has only two solution.

Since,         

or   by definition of prime

or  

Combining both, we can write it as:

  

Therefore,   

   we got our original congrunce relation. Hence, - 1,+1 are only to solutions which are integer.  

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