On this submit we discover all Pythagorean triples that comprise consecutive numbers, all Pythagorean triples (a, b, c) such that a + 1 = b or b + 1 = c.
a + 1 = b
George Osborne wrote a paper [1] addressing the query of when the squares of two consecutive numbers can be a sq.. Geometrically that is asking for primitive Pythagorean triples for which the legs are consecutive integers.
He proved that the sequence shorter legs satisfies the recurrence relation
with preliminary situations u0 = 0 and u1 = 1. That is OEIS sequence A001652.
The strategy for fixing recurrences just like the one above is analogous to the tactic for fixing linear differential equations. See an answer right here. This provides us the next method for the phrases:
b + 1 = c
It’s additionally attainable for the longer aspect and hypotenuse of a Pythagorean triangle to be consecutive numbers, as within the (5, 12, 13) triangle.
All primitive Pythagorean triples are given by Euclid’s method
with integers m > n > 0. If b and c are consecutive numbers, then
and so m = n + 1. Due to this fact all attainable values of b are given by 2n(n + 1) for n > 1.
[1] Geo. A. Osborne. A Downside in Quantity Idea. The American Mathematical Month-to-month, Vol. 21, No. 5 (Might, 1914), pp. 148-150
