3
m), then the is a Fibonacc -r is a Give both cases. CI NUMBERS are somewhat sparse; only 42 of them are being the 126377-digit u604711. present Pr, arranged in ascending order. Next, consider the correspond- U5, U p, . According to Theorem 14.3, these are 1. Each of the remaining .. primes 2, 3, 5. r - 1 numbers is %3D Up-1 or u p+1 divisible by p. Confirm PROBLEMS 14.2 u-). is in the cases of the primes 7, 11, 13, and 17. n+1 %3D ST are all even integers. are all divisible by 5. Prove that if 2 |Un, then 4 | (un+1 -Un-1); and similarly, if 3 | un, then 9 | (u²1 – u? ,). (a) un+3 = Un (mod 2), hence u3, u6, U9, (b) un15 = 3un (mod 5), hence u5, u10, u15, E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula 3. Prove that if 2|Un, then 4| (u? .. %3D Itun"n = "n+ ..+ En + Gn + ;n [Hint: For n > 2, u = unun+1 - unun-1.] 6. Utilize the identity in Problem 5 to prove that for n > 3 = u+ 3u-1 + 2(u, 2 + ude %3D 7. Evaluate gcd(u9, u12), gcd(u15, u20), and gcd(u24, U 36). 8. Find the Fibonacci numbers that divide both u24 and u 36. u= u;+3u-1 +2(u-2 + u-3 + ·..+ už + uf) . Use the fact that u m lu, if and only if m | n to verify each of the assertions below: (a) 2|un if and only if 3 | n. Un+1 arize 3. = (21 (1) (b) 3| un if and only if 4 | n. (c) 5| un if and only if 5 | n. (d) 8| un if and only if 6 | n. %3D for all m, n > 1. divides umn %3D number or u m Is a Fibonacci number. Give examples illustrating both cases. numbers. Find them. %3D Hin , prove that 2"-lun =n (mod 5). a Fibon "/>
Extracted text: (Hint: Use fact that 2" = + 4(2"-2Un-1).] 12. It was in 1989 that there are only five are also triangul= 11. It can be that when un is divided by um (n > m), then the is a Fibonacc -r is a Give both cases. CI NUMBERS are somewhat sparse; only 42 of them are being the 126377-digit u604711. present Pr, arranged in ascending order. Next, consider the correspond- U5, U p, . According to Theorem 14.3, these are 1. Each of the remaining .. primes 2, 3, 5. r - 1 numbers is %3D Up-1 or u p+1 divisible by p. Confirm PROBLEMS 14.2 u-). is in the cases of the primes 7, 11, 13, and 17. n+1 %3D ST are all even integers. are all divisible by 5. Prove that if 2 |Un, then 4 | (un+1 -Un-1); and similarly, if 3 | un, then 9 | (u²1 – u? ,). (a) un+3 = Un (mod 2), hence u3, u6, U9, (b) un15 = 3un (mod 5), hence u5, u10, u15, E Show that the sum of the squares of the first n Fibonacci numbers is given by the formula 3. Prove that if 2|Un, then 4| (u? .. %3D Itun"n = "n+ ..+ En + Gn + ;n [Hint: For n > 2, u = unun+1 - unun-1.] 6. Utilize the identity in Problem 5 to prove that for n > 3 = u+ 3u-1 + 2(u, 2 + ude %3D 7. Evaluate gcd(u9, u12), gcd(u15, u20), and gcd(u24, U 36). 8. Find the Fibonacci numbers that divide both u24 and u 36. u= u;+3u-1 +2(u-2 + u-3 + ·..+ už + uf) . Use the fact that u m lu, if and only if m | n to verify each of the assertions below: (a) 2|un if and only if 3 | n. Un+1 arize 3. = (21 (1) (b) 3| un if and only if 4 | n. (c) 5| un if and only if 5 | n. (d) 8| un if and only if 6 | n. %3D for all m, n > 1. divides umn %3D number or u m Is a Fibonacci number. Give examples illustrating both cases. numbers. Find them. %3D Hin , prove that 2"-lun =n (mod 5). a Fibon