Let  Prove that for every positive integer n the numbers x_n, y_n, and z_n are the lengths of the sides of a right triangle and that x_n and y_n are consecutive integers.

28. Discover and prove properties of the Pell sequence that are analogous to those of the Fibonacci sequence.

29. Let the sequence 1, 5, 85, 21845, ... be defined by
 

30. Let a sequence be defined by

Show that d_n = 3c_n + 1, where c_n is as defined in the previous problem.

31. Prove that 

^*32. Certain of the above formulas suggest the following:

Prove it for general m.

^*33. Prove that n⁵ - n is an integral multiple of 30 for all integers n.

^*34. Prove that n⁷ - n is an integral multiple of 42 for all integers n.

^*35. Show that every integer from 1 to 2ⁿ⁺¹ - 1 is expressible uniquely as a sum of distinct powers of 2 chosen from 1, 2, 2², ... , 2ⁿ.

^*36. Show that every integer s from has a unique expression of the form

where each of c₀, c₁, ..., c_n is 0, 1, or -1.
 
 

Saturday, April 11, 1998