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.
Cheers,
Nalin Pithwa.