I have to determine $$13^{498} \pmod{997}$$
I know that it can only be $1$ or $-1$. But I don't quite know which. How can I decide?
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Do you mean 498 instead of 488 as that is 1 or -1 or maybe even 977 instead of 997 – John Marty May 12 '15 at 08:52
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@JohnMarty- I'm sorry. I mean 498 and 997 – algebraically_speaking May 12 '15 at 08:55
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1@ Dietrich Burde - I think this is different as it is very specifically looking at what $x^{p-1/2}$ can be. – John Marty May 12 '15 at 08:59
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Use Euler's criterion and quadratic reciprocity to find $$ 13^{498}\equiv\left(\frac{13}{997}\right)=\left(\frac{997}{13}\right)=\left(\frac9{13}\right)=1 $$
Martin Sleziak
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