Let p be a prime larger than 3. 1. Show that the remainder, when p is divided by 6, is 1 or 5. Note: when a positive integer is divided by n > 1, the possible remainders are 0, 1, 2, ... , n – 1. 2. Show that 12 | p² – -1.

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Q: suppose a prime p of the form 8n + 3 and that q = (p-1) / 2 is also prime. Prove that ordp (2) = 8n…

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Q: 2. Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? +1 is divisible by 10. 3.…

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Q: Let p25 be a prime. Prove that p(p+1) (p-1)! +p+1.

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Q: Show that if p is an odd prime, then n = 2º – 1 passes Miller's test base 2 -

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Q: Let p be an odd prime not equal to 5. Prove that either p² – 1 or p? + 1 is divisible by 10.

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Q: (1) Prove that is a, b, and c are integers with c + 0 then ac = bc = a = b ( that is this a key step…

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Q: 6. (i). Prove that every prime > 3 is of the form 6m +1 or 6m + 5.

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Q: For p an odd prime in Z , Prove: If -1 is a perfect square in Z/pZ then p= 1(mod 4)

A: This can be done as follows:

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Q: For p an odd prime in Z , Prove: If p=(a)^2 + (b)^2  is possible in Z then   p is not prime in Z[i]

A: Given that p is odd prime. This implies that p is not equal to 2.

Q: ) Show that if p is a prime such that m +1< p < 2m + 1, then p divides (2m+1).

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Q: السؤال 2 For all n E N, n> 1, there exists a prime p such that p|n. ylan O İhi O

A: True

Q: Show that for any odd prime p, the Gauss, G(p), can be rewritten as p-1 G(p) = 5². x=0

A: To Show- Show that for any odd prime p, the Gauss, Gp, can be rewritten as Gp = ∑x=0p-1ζpx2.

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Q: Find all prie 8. If p > q 2 5 and p and q are both primes, prove that 24 p2-q?.

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A: Let σp2q=σp2σq Then we have  σp2q=σp2σq=1+p+p21+q=p3-1p-1q2-1q-1≤p3p-1q2q-1

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A: Solution:-

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Q: Let p ≥ 5 be a prime. Prove that p(p+1) | (p − 1)! +p+1. -

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Polynomials

Try substituting 1. You will get 6 and not 0. Try other real values also. What About -2? Let us check.