# 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

*c*=

*b*+

*1,*

*a*,

*b*,

*c*is a Pythagorean triplet; that is, such that

*a*

^{2}+ b^{2}= c^{2}?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,

*a*) but

^{2}+ b^{2}= c^{2}*c*≠

*b*+

*1 ?*

Indeed, is it possible to derive a formula defining an entire series of such numbers

*b*... from each of which

_{1}, b_{2}, b_{3}, b_{4},*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.