2 biquadratic residue mod p

  • Thread starter b0mb0nika
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  • #1
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show that x^4 == 2( mod p) has a solution for p==1(mod 4) iff
p is of the form A^2+64B^2, where A,B are integers


I let x^2=M
then the conguence is reduced to M^2==2( mod p)
but any # squared == 0 or 1 ( mod4 ) so p must be == 1(mod4)...
but i'm not sure what to do now..
any hints/ or ideas ?
 

Answers and Replies

  • #2
matt grime
Science Advisor
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Find when 2 is a residue mod p using, say, Quadratic Reciprocity, and then find when its square root is also residue, again using reciprocity
 

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