Number Theory

Find all solutions in integers y and z to the equation 62 + y2 = z2.

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Let a and b be positive integers. Show that (ab)(gcd(a; b)) = (a)(b) gcd(a; b). (Example: If a = 12 and b = 8, the equation reads 32 2 = 4 4 4.)

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Prove that there are an innite number of primes congruent to 3 mod 4.

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"Fermat showed that 237 -1 is composite by nding a prime factor p of 237 -1 which lies between 200 and 300. Using your knowledge of number theory, deduce the value of p."

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"Which positive integers m have the property that there is a primitive root mod m? (Summarize what we know about this question, and why we know it. Your answer should be clear enough that one could use it to decide immediately if there is a primitive root modulo (257)2, 4 661, 257 661, : : : .)"

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Find the number of solutions to the congruence x2 9 mod 23 112.

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"Find a positive integer n such that n=3 is a perfect cube, n=4 is a perfect fourth power, and n=5 is a perfect fth power."

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Prove that there are no whole number solutions to the equation x2 - 15y2 = 31.

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Find the number of solutions to the congruence x2 9 mod 23 112.

Hello! I am a student of Masters. I have to submit my thesis on this topic so I need your help in preparing it. Kindly help me in this situation. I wi...

"Which positive integers m have the property that there is a primitive root mod m? (Summarize what we know about this question, and why we know it. Your answer should be clear enough that one could use it to decide immediately if there is a primitive root modulo (257)2, 4 661, 257 661, : : : .)"

Hi! I am appearing for the entry test of a local top ranking university and need your help for preparations. Can you suggest a worthy answer to my que...

"Fermat showed that 237 -1 is composite by nding a prime factor p of 237 -1 which lies between 200 and 300. Using your knowledge of number theory, deduce the value of p."

Hey! I wanted to ask whether my question has any reasonable answers. I am doing my home assignment and I’m not quite sure about the answer I have in m...

Prove that there are an innite number of primes congruent to 3 mod 4.

We have formed a team in our college council where we provide resources to our subscribers related to exam material. I personally thought of including...