Question

Find the value of the Legendre symbol (−2/p), depending on the congruence class of p modulo 8.

Answer #1

determine the value of Legendre symbol (-8/47)

Prove: If p is prime and p ≡ 7 (mod 8), then p |
2(p−1)/2 − 1. (Hint: Use the Legendre symbol (2/p) and
Euler's criterion.)

Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Number Theory:
Find a square root of −1 modulo p for each of the primes p = 17
and p = 29. Does −1 have a square root modulo 19? Why or why
not?

Given that 2 is a primitive root modulo 19, find all the
primitive roots modulo, 19. You must know how you are getting your
answer and make sure all your answers are in the canonical residue
set

find the least positive residue modulo 51 of the following integer
2^1282.

Write the symbol for the present value of an annuity immediate
that pays $2 monthly at an annual effective rate of 8% for 10 years
then calculate its value.

(i) Verify that 2 is a primitive root modulo 29.
(ii) Find all the primitive roots modulo 29. Explain how you
know you have found them all.
(iii) Find all the incongruent solutions to x6 ≡
5(mod 29).

6. What is the orbit of 2 in the group Z_7 under multiplication
modulo 7? Is 2 a generator?
7. What is the residue of 101101 modulo 1101 using these as
representations of polynomials with binary coefficients?
8. List all irreducible polynomials with binary coefficients of
degree 4 or less. (Hint: produce a times table that shows the
minimum number of products needed.) Show these as binary numbers
(omitting the indeterminant) and as decimal numbers (interpreting
the binary number into...

Find all solutions to each congruence.
(a) 2x − 3 ≡ 2 (mod 7)
(b) 3x + 4 ≡ 1 (mod 5)
(c) 3x ≡ 6 (mod 9)
(d) 14x ≡ 11 (mod 15)

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