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I believe all right-angled triangles with integer sides can be found
using:
a = k*2xy b = k (x^2 - y^2) c = k (x^2 + y^2)
where k, x, and y are positive integers, and x > y.
And GCD (x,y)=1
If GCD(a,b,c)=1 then k=1 so the semi-perimeter
S = 1/2 (2xy + x^2 - y^2 + x^2 + y^2)
= x (x + y)
Now if b and c are to be consecutive integers...
c - b = 1
x^2 + y^2 - x^2 + y^2 = 1
2y^2 = 1 which has no integral solution.
This should have been more obvious to me since the longest
side and shortest side cannot possibly be consecutive!
But if a and c (the longest two sides) are consecutive integers...
c - a = 1
x^2 + y^2 - 2xy = 1
(x-y)^2 = 1
x - y = +1 which means y must be (x - 1) with x>=2
(x and y are consecutive! not to be confused with a,b,c)
Finally, the semi-perimeter
S = x (x + y)
= x (x + x - 1)
= x (2x - 1)
A triangle number is T(n) = SUM [i], i=1,n = n(n+1)/2
Let n = 2x - 1 x>=2
Then T(n) = (2x-1)(2x)/2 = x(2x-1) = S
So the semi-perimeter starts with T(3) and every other one thereafter.
("odd" triangle numbers)
E.g.
x=2 n=3 yields triangle (4,3,5) S=6
x=3 n=5 yields triangle (12,5,13) S=15
x=4 n=7 yields triangle (24,7,25) S=28
x=5 n=9 yields triangle (50,9,49) S=54
etc.
It appears that in a triangle defined by .0, the semi-perimeter is
a triangular number iff the two longer sides are consecutive integers.
Symmetry explains the "either... or" clause in .0
I've only proved half of the iff, however. i.e. if consec then
triangle.
To prove the converse: if S is triangle number then consec is not so
obvious...
For odd triangle numbers, T(odd), it is trivial to show that
x - y must be 1 and therefore c - a = 1. Note that x and y are
consecutive if and only if c and a are consecutive.
For T(even), say T(2x) without loss of generality, I try to
solve T(2x)=x(x+y)
2x(2x+1)/2 = x(x+y)
2x+1 = x + y
x = y - 1
This contradicts x>y as stated at the beginning. So this implies that
that the S of a right triangle, if a triangle number, must
be an odd triangle number.
Therefore, "if S is triangle then consec" is true.
And therefore it is *PROVEN* that in a right-angled triangle, the
semi-perimeter is a triangular number if and only if the two longer
sides are consecutive integers.
Allen
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