Supernatural Triangles

A Pythagorean triangle is called supernatural if two of its three sides are consecutive
integers.

Let $S(N)$ be the sum of the perimeters of all distinct supernatural triangles with perimeters less than or equal to $N$.
For example, $S(100) = 258$ and $S(10000) = 172004$.

Find $S(10^{10^{10}})$. Give your answer modulo $1234567891$.

This is a complex problem in the field of number theory. To understand it, let’s break it down piece by piece.

Firstly, a Pythagorean triangle refers to a right-angled triangle where the lengths of the three sides are integers that satisfy the Pythagorean theorem: a² + b² = c².

A “supernatural” Pythagorean triangle specified by the problem is a special kind of Pythagorean triangle where two of its sides are consecutive integers.

More specifically, if we call the consecutive integers a and a+1, we can deduce that the third side, as a result of the Pythagorean theorem, would be √((a+1)² + a²). To maintain the integrity of Pythagorean triangles, we can infer that the √((a+1)² + a²) must be an integer, therefore (a+1)² + a² should be a perfect square number.

Thus, we are looking for pairs of consecutive perfect squares. This isn’t trivial, but it can be shown that the only such pairs of squares are those of the form {n^2, (n+1)^2} where n is a Fibonacci number.

The Fibonacci sequence is well-known: F(n+2) = F(n+1) + F(n), with F(1) = F(2) = 1.

This connection gives the problem a solution in terms of Fibonacci numbers.

Before going into the fine details, let’s simplify the problem by noting that we’re looking for triangles with a perimeter less than N. This means 2*a + √((a+1)² + a²) <= N. As the perimeter of the triangle increases as 'a' increases, we can solve the inequality for 'a' to give us the maximum 'a' which we can then use to generate the required triangles and sum the perimeters. The sum of the perimeters under a given limit is the sum of F(n)*F(n+2) where n ranges over values such that F(n+2) <= perimeter limit. The sum of products of Fibonacci sequences has been solved in multiple research papers involving Fibonacci sequences or inequalities. The final answer can be computed directly using the Binet's formula for Fibonacci numbers and summing up the sequence that falls within the limit stated. Since the sum S(10^{10^{10}}) is extremely large, it is required to present it as a modular number (modulo 1234567891). Computing such a large numbers involves advanced mathematical techniques and high precision softwares or programming languages that could handle such number. Considering the magnitude of this problem, this is often considered an out of reach problem for ordinary mathematics, and covers the topics that are studied in Number theory, Fibonacci sequence, and modular arithmetic. In essence, to solve this problem, we need to find all Fibonacci numbers within the limit, square them, find the sum of the sequence, and finally present it modulo 1234567891. Important Note: This problem is considered an extremely high level problem (even beyond the level of a common Mathematics Olympiad). It's also possible that it was designed as a problem not to be straightforwardly solvable due to the massive number involved (A googol, which 10^{10^{10}} represents, is incredibly large). Even with a powerful computer, in practical terms, the calculation might take more than the age of the Universe!

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