Fun with mathamatics

More properties of Pythagorean triples

It is found that cⁿ  - a ⁿ   - bⁿ    is always divisible by (n/2)^{3  ,} where n

is even and (a,b,c)  are any Pythagorean triples.

It is found that cⁿ  - a ⁿ   - bⁿ     is  divisible by a² and  b²    for all

even values of n .>2

e.g., c⁴  - a⁴  - b⁴  = 2a² b²  and c⁶ – a⁶ – b⁶  = 3a² b² c² 

All the members of a Pythagorean triple cannot be odd, since sum

of two odd squares is even which cannot be equal to square of the

other odd number.

It implies that all the members of any triple cannot be prime.

All of them can be even. Such Pythagorean triples will always be

reducible.

All of them cannot be square numbers, cube numbers or in general

equal power numbers with power n >1. If (a² , b² , c² ) exists as a

 possible set of Pythagorean triple, then the Pythagorean relation

demands that  a⁴  + b⁴  =  c⁴  which is not possible as it is against

the Fermat’s last  theorem.

1 cannot be a member in any Pythagorean triples. If 1 is a member

( lowest) then 1 =  c²  - b²   or 1 = (c+b)(c-b). The mathematical

description says that the product of a number and its reciprocal will

always give unity. Since c-b cannot be a fraction and  c² – 1 cannot

be a square number , for any integral values of c and b  the product

cannot be made to unity.