Primitive Pythagorean Triples which form an Arithmetic Progression.
Let be a primitive Pythagorean triplet , and they also follows an arithmetic progression .
Find the sum of the possible values of .
The answer is 12.
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1 solution
By definition, If three positive numbers are in Arithmetic Progression , they can be written as a-d, a and a+d, where a and d are greater than zero.
By our definition, if these three numbers are a Pythagorean triple then
(a-d)^2 + a^2 = (a+d)^2
Expanding the terms and simplifying we get
a^2 = 4 a d
Since a > 0, we can divide both sides by a and get
a = 4d
So d is a divisor of a,
Since a and d are supposed to have no common divisor other than 1 it means d = 1 ie.
a = 4,
a - d = 3
and
a + d = 5
Therefore (3,4,5) is the ONLY primitive Pythagorean triple where the three numbers are also in Arithmetic Progression...