Construct a polynomial equation x7 a1x6 a2x5 a3x4 a4x3

Question

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...F_n-1 = F_n - 2, via induction on n.

7. Show that F_n:= 2²ⁿ + 1 is prime if and only if:  5^(Fn+1) ≡ -1 mod F_n.

8. Show that F₆ := 2²⁶ + 1 is composite via Pepin's test.

9. Evaluate: 59^9258008 mod 11371601, via two different methods. Observe that 11371601 = 59.97.1987.

10. Solve: x^1536307 = 34291849, via two different methods. Notice that 34291849 = 5233.6553.

Answer 1

The following assignment is based on polynomial equation 1. Construct a polynomial equation: x7 + a1x6 + a2x5 + a3x4 + a4x3 +a5x2 + a6x + a7 = 0 mod 7, with exactly 4 roots. You must identify 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. Illustrate that the Mersenne number: M607: = 2607 - 1, is a prime via Lucas-Lehmer Algorithm. 4. Find the prime features of the Mersenne number: M233: = 2233 - 1, via Euler's Theorem. 5. Show that n is prime if and only F(n) = n - 1. 6. let Fn := 22n + 1 be Fermat number Show that: F0F1...Fn-1 = Fn - 2, via induction on n. 7. Demonstrate that Fn:= 22n + 1 is prime if and only if: 5(Fn+1) = -1 mod Fn. 8. Prove that F6 := 226 + 1 is composite via Pepin's test.