Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of Fp. For which of the following primes is 3 a primitive root modulo p? (i) p = 5 (ii) p = 7 (iii) p = 11 (iv) p = 17

Q: Let p + 7 be an odd prime. Determine the value of the Legendre symbol (7/p).

A: Let p be an odd prime number. An integer a is a quadratic residue modulo p if it is congruent to a…

Q: s 19 a prime in the Eisenstein integers? is 79? If they are, explain why, if not, display a…

A: Given: Primes in Eisenstein integer. To find: a) To check wheater 19 and 79 are primes in Eisenstein…

Q: Can you please show me step by step for this math that makes me undertandable. Q: Determine which…

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Q: Show that for every integer n, either 3 | n or n² = 1 (mod 3).

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Q: 12.16. Give an example of a prime p of each of the forms: a. 4k + 1 b. 4k + 3

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Q: 7. Prove that if n is an odd positive integer or divisible by 4, then 13 +23 + 33 + .+ (n– 1)3 = 0…

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Q: all Natural numbers, n, that make 62 ≡≡ 46 mod n. Show all your work:

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Q: let n be an integer solution to the equation n2+an-b=0 (n>7) if the representation of a in base n is…

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Q: Show that if p = q + 4a (p and q are odd primes), then (p/q) = (a/q).

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Q: Podasip 11. If p is prime, then every c can be expressed as a sum of two squares: x² + y² = c(mod…

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A: Let n be any integer. Then n3-n is divisible by 3.

Q: p, q, pq | (apq − ap − aq  + a) for all a.

A:

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: na ed t 4. Use the theory of congruences to verify that 25| (2"+4 + 33n+2 – 53n+6) for all n eN

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Q: (b) Find a prime p> 5 such that 2 and 3 are both primitive roots modulo P.

A: We test all the primes p>5 and see which prime works.

Q: If n is a positive integer, is there always a prime between n and 2n?

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Q: Suppose that a, b are integers such that 1 = -19a + 45b. Use this information to find all integers a…

A: The given equation 1=−19a+45b suggests that 1 is a linear comb.−19 and 45 with coefficients a and…