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 have, as far as I know, not been published anywhere else before.
Given any odd integer a, derive a (larger, even) integer b such that,
if c = b + 1,
a, b, c is a Pythagorean triplet;
that is, 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 c ≠ 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 thee questions. However, I suspect that the answer is that for many odd integers n
there exists no Pythagorean triplet in which n appears.