295 For any primep 2 or 5, it is known that either Fp-1 or Fp+1 is divisible by p. Confirm this in the case of the primes 7, 11, 13, and 17. 3. From the formula F+ F- F2 (-1)", conclude that consecutive Fibonacci numbers are relatively prime. 4. n One can prove that the greatest common divisor of two Fibonacci numbers is also a Fibonacci number; 5. specifically, where d gcd (n, m). gcd (F1, Fm)= Fd no ner. Verify this identity in the case of gcd(F9, F12) and gcd(F15, F20). mula Use Problem 5 to prove that for n > 2, Fn| Fm if and only if n m. 6. no2 7. Establish each of the following assertions: 2|F, (that is, F, is even) if and only if 3 n. (b) 31F, if and only if 4|n. (c) 4|F, if and only if 6 n. (d) 5F, if and only if 5 n. [Hint: All these require the aid of the previous problem.] 8. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula ing F2+F+F. .+ F= F,F+ 1 [Hint: Note that F F,(Fn+1 - F1-1) F,F +1- FrFn-1.]

295 For any primep 2 or 5, it is known that either Fp-1 or Fp+1 is divisible by p. Confirm this in the case of the primes 7, 11, 13, and 17. 3. From the formula F+ F- F2 (-1)", conclude that consecutive Fibonacci numbers are relatively prime. 4. n One can prove that the greatest common divisor of two Fibonacci numbers is also a Fibonacci number; 5. specifically, where d gcd (n, m). gcd (F1, Fm)= Fd no ner. Verify this identity in the case of gcd(F9, F12) and gcd(F15, F20). mula Use Problem 5 to prove that for n > 2, Fn| Fm if and only if n m. 6. no2 7. Establish each of the following assertions: 2|F, (that is, F, is even) if and only if 3 n. (b) 31F, if and only if 4|n. (c) 4|F, if and only if 6 n. (d) 5F, if and only if 5 n. [Hint: All these require the aid of the previous problem.] 8. Show that the sum of the squares of the first n Fibonacci numbers is given by the formula ing F2+F+F. .+ F= F,F+ 1 [Hint: Note that F F,(Fn+1 - F1-1) F,F +1- FrFn-1.]

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.2: Exponents And Radicals

Problem 92E

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