Reference no: EM13917137
1. Construct a polynomial equation:
x7 + a1x6 + a2x5 + a3x4 + a4x3 +a5x2 + a6x + a7 ≡ 0 mod 7, with exactly 4 roots. You must specify those 4 roots.
2. When a farmer counted his eggs by 4's, there was 3 left over, when counted by 5's, there was 3 left over, when counted by 6's, there was 1 left over, when counted by 11's, there was 8 left over. What is the smallest number of eggs the farmer had?
3. Show that the Mersenne number: M607 := 2607 - 1,is a prime via Lucas-Lehmer Algorithm.
4. Find the prime factors of the Mersenne number: M233 := 2233 - 1, via Euler's Theorem.
5. Show that n is prime if and only Φ(n) = n - 1.
6. let Fn := 22n + 1 be Fermat number Show that:
F0F1...Fn-1 = Fn - 2, via induction on n.
7. Show that Fn:= 22n + 1 is prime if and only if: 5(Fn+1) ≡ -1 mod Fn.
8. Show that F6 := 226 + 1 is composite via Pepin's test.
9. Evaluate: 599258008 mod 11371601, via two different methods. Observe that 11371601 = 59.97.1987.
10. Solve: x1536307 = 34291849, via two different methods. Notice that 34291849 = 5233.6553.
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Construct a polynomial equation
: Construct a polynomial equation: x7 + a1x6 + a2x5 + a3x4 + a4x3 +a5x2 + a6x + a7 ≡ 0 mod 7, with exactly 4 roots. You must specify those 4 roots
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