More fun in 2015, Part 32
How many of the statements (A) and (B) below are true?
(A) : There exists a right triangle with three integer sides whose area is 2015.
(B) : There exists a right triangle with three rational sides whose area is 2015.
Dedicated to Mr. Mendrin , who insisted that I post this very challenging problem.
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1 solution
We can easily see why statement (A) is false. For an integer-sided right triangle with legs a and b to have area 2 0 1 5 , we must have 2 1 a b = 2 0 1 5 ⟹ a b = 4 0 3 0 . Simply list the pairs of factors of 4 0 3 0 and check to see that none of them are legs of a Pythagorean triple.
Statement (B) is much more complicated. It is equivalent to the statement " 2 0 1 5 is a congruent number ," and the general problem of determining whether a given integer is congruent has been open for a long time. Other than pointing out that 2 0 1 5 is on this list of congruent numbers on OEIS, I can't give a very good explanation for why statement (B) is true. Maybe someone with more knowledge of algebraic number theory can provide more insight.