1. A simplified form of Fermat’s theorem: If x, y, z, n are natural numbers, and , prove that the relation does not hold.
2. Distribution of numbers: Find ten numbers such that (a) the number is contained in the closed interval (b) the numbers and lie in different halves of the closed interval (c) the numbers , , lie in different thirds of the closed interval (d) the numbers , , and lie in different quarters of the closed interval , etc., and finally, (e) the numbers , , lie in different tenths of the closed interval
3. Is generalization of the above possible?
4. Proportions: Consider numbers A, B, C, p, q, r such that: , , , write the proportion in such a way that in the empty squares, there will appear expressions containing p, q, r only; these expressions being obtained by cyclic permutation of one another expressions.
5. Give an elementary proof of the fact that the positive root of is irrational.
I will give you sufficient time to try these. Later, I will post the solutions.