# Mathematical puzzles

Mathematical puzzles can perhaps always be akin to school homework or challenges to prove theorems in the style of the Elements of Euclid; still, they are more unusual than the school homework tasks the aforementioned represent. The following has not, as far as I know, been published anywhere else before except in SEMantics, the regional newsletter for the south east England of British Mensa.

#### Pythagorean triplets

Given any odd integer a, is it possible to derive a (larger, even) integer b such that, if = b + 1, a, b, c is a Pythagorean triplet; that is, such that a2 + b2 = c2 ?

Given any odd integer a, is it possible to derive (larger) integers b and c such that a, b, c is a Pythagorean triplet (that is, a2 + b2 = c2 ) but  b + 1 ?

Indeed, is it possible to derive a formula defining an entire series of such numbers b1, b2, b3, b4, ... from each of which c can be determined and is also still an integer?

Note: I do not have a formula as an answer, or indeed any answer (as to whether such a formula exists) for either of these questions. However, I suspect that the answer is that for many odd integers n there exists no Pythagorean triplet in which n appears.