can never divide?

1. Given that x is a positive integer prove that f(x) = x² + x + 1 will never divide by 5.

2. Consider the expression x^x + 1, where x be a positive integer.

It can be verified that x = 7 is the least value for which x^x + 1 divides by 2³.

Given that n is a positive integer, find the least value of x for which x^x + 1 is divisible by 2ⁿ.

convolutions and sum of sums

Let S_k be the sequence of simple sums, that is S₁=1, S₂ =1+2, S_k =1+2+...+k then

S₁ + S₂ + ... + S_k = k.1+(k-1).2+(k-2).3+ ... + 2.(k-1)+1.k = the convolution of the sequence {1, ...,k}.

In your spare time you can show that sum of sums equals (n)(n+1)(n+2)/(1.2.3) and sum of sum of sums equals n(n+1)(n+2)(n+3)/(1.2.3.4) and so on.

1.2.3 = 6.
1.2.3.4 = 24.

Fibonacci Sums

The Fibonacci series is: F₁=1, F₂=2, for n=2,3,... F_n+1= F_n + F_n-1. The (n+1)-st number is the sum of n-th and (n-1)-th number in the sequence.

Fibonacci numbers have the property that: the sum of the first n numbers of a sequence is contained in the sequence. Do you know of others?

F₁+F₂+F₃+...+F_n = F_n+2-1

Actually the sequence G₁=1, G₂=2,for n=2,3,... G_n+1= G_n + ... + G₂ + G₁ trivially staifies that property. So, Fibonacci sequence is not unique in the above sense. Can you think of a sequence {H_k} such that

H₁+H₂+H₃+...+H_n = H_n+3 - c? for some constant c.

On a tangent: Kolmogorov information complexity speaks of representations that can compress information effectively. Not only does the following set of characters "F₁=1, F₂=2 F_n+1= F_n + F_n-1" contain the entire Fibonacci sequence, but also the sum of its first n elements.