Prove or disprove the following statements:(a) If n e Z then the equation n³ + (n + 1)³ + (n + 2)³ = (n + 3)³ has a uniqueintegral solution.(b) If n = a² for some a E Z then 1155 n"" + 7777n + 1(c) (n² + n+ 41)² = 1 mod 12 , for every positive integer n%3D(d) (11 × 14" + 1) is a composite integer for every positive integer n

Since you have asked multiple question, we will solve the first question for you. If you want any…

Prove or disprove the following statements: (a) If n E Z then the equation n3 + (n + 1)³ + (n + 2)³ = (n + 3)³ has a unique integral solution. (b) If n = a² for some a e Z then 1155 n77 + 7777n +1 (c) (n² + n+ 41)² = 1_mod 12 , for every positive integer n (d) (11 × 14" +1) is a composite integer for every positive integer n

Prove or disprove the following statements: (a) If n E Z then the equation n3 + (n + 1)³ + (n + 2)³ = (n + 3)³ has a unique integral solution. (b) If n = a² for some a e Z then 1155 n77 + 7777n +1 (c) (n² + n+ 41)² = 1_mod 12 , for every positive integer n (d) (11 × 14" +1) is a composite integer for every positive integer n

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

2. p²-1 (²³) = (-1) ² 8 1 {} -1 || if p= 1 (mod 8) if p= ±3 (mod 8).
2. Prove by induction that Er² = n(n+1)(2n+1). Deduce formulae for 1.1+2·3+3.5+4.7+...+n(2n-1) and 12+3² +5² +... (2n-1)².
n>1 Problem 4. Prove by induction that, for every geq2 (1-)(1--)-(1-)- n +1 1 - 32 42 n2 2n
1.12 Show that (n+1)4 n for all n ≥ 5.
Prove by induction that E1 (32') = n2gn+1 n27+2 + 3- 2+1-6 whenever n is a positive integer
222
5. Use ordinary induction to prove that for every positive integer n, n-n is a multiple of 6. Only proofs by induction are accepted.
7. Suppose n is a natural number that is greater than 100. Alex and Jerry each compute the gcd(4n. 20n2 + 15) and obtain different answers. Alex's computation has 12 divisions and Jerry's has 19. (a) Can we conclude that Alex's calculation contains an error? (b) Can we conclude that Jerry's calculation contains an error?
7. Use mathematical induction to show that п Σ 1 4i (4n+1 – 1) 3 i=0 for all integers n2 0.
Suppose gcd(a,n) = 1. Then aφ(n) ≡ 1 (mod n).