Pythagorean triples by trial and error

This simple C program finds primitive Pythagorean triples by trial and error (trying each potential value to determine whether or not it is a valid triple). This is a slow way to find such triples.

C code

/* Find primitive Pythagorean triples by trial and error

Version 1.4 (2016-09-30, revised 2019-03-04)

We want positive integers a <= b <= c such that a squared + b squared = c squared.

We only want primitive triples, so a, b, and c cannot have a common factor. If any two of a, b, and c have a common factor, the other one of them will have that common factor and so we need to make sure that no two of them share a common factor. No two of a, b, and c are equal and at most one of them is even. If at most one of a, b, and c is even, at least two of them are odd. If two of them are odd, the other must be even, because the square of an odd number is odd and the sum of two odd numbers is even and the difference between two odd numbers is even. So, only one of a, b, and c is even.

If a and b were odd and c was even, we could find integers i, j, and k such that a = 2i + 1, b = 2j + 1, and c = 2k. We could write a squared + b squared = c squared as (2i + 1) squared + (2j + 1) squared = (2k) squared, which we can write as 4 i squared + 4i + 1 + 4 j squared + 4 j + 1 = 4 k squared or 4 (i squared + j squared - k squared + i + J) + 2 = 0 or 2 (i squared + j squared - k squared + i + j) + 1 = 0. No integers i, j, and k can satisfy this equation, because the left-hand side is an odd number, but the right-hand side is even. So, c cannot be even.

So, in a primitive triple, one of a and b is even and c is always odd.

If the value a or b that was even was the product of 2 and an odd number, we could write it as 2(2i + 1) and we could write the other two values of the triple as 2j + 1 and 2k + 1. We could write a squared + b squared = c squared as 4 (4 i squared + 4 i + 1) + 4 j squared + 4 j + 1 = 4 k squared + 4 k + 1 or 4 (i squared + i) + 1 + j squared + j) - (k squared + k) = 0. Regardless of whether j or k are odd or even, j squared + j and k squared + k are always even. No integers i, j, and k can satisfy this equation, because the left-hand side is an odd number, but the right-hand side is even. So, the even integer a or b is not the product of 2 and an odd number. So, when a or b is even, it must be divisible by 4.

We know that b < c. The maximum possible value of b is (c-1). When b = c - 1, c = b + 1 and we can use a squared + b squared = c squared to give a squared + b squared = b squared + 2 b + 1 or 2 b = a squared - 1.

When a is even, b and c are odd, so the maximum value of b is (c-2). When b = c - 2, c = b + 2 and we can use a squared + b squared = c squared to give a squared + b squared = b squared + 4 b + 4 or b = (a / 2) squared - 1.

If a is odd, a must be the product of two odd numbers, one of which is the value for a of a primitive triple. The minimum odd value for a is 3, so the largest common factor of the triple that we must test when a is odd and greater than 3 is a / 3.

If b is odd, b must be the product of two odd numbers, one of which is the value for b of a primitive triple. This primitive triple has an odd value for b, so has an even value for a. The minimum even value for a is 8, so the largest common factor of the triple that we must test when b is odd and greater than 8 is a / 8. */

/* Standard header file for fprintf, fputs, printf, puts */

#include <stdio.h>

/* Standard header file for EXIT_FAILURE, EXIT_SUCCESS, free, malloc, NULL, realloc */

#include <stdlib.h>

/* Standard header file for strcmp */

#include <string.h>

/* Standard header file for sqrt */

#include <math.h>

/* Returns the position of a character in a character array */

unsigned char position_in_character_array (char const character)

/* Returns the maximum position of a character in a character array across all characters in a string */

unsigned char maximum_position_in_character_array (char const * const string)

/* Sets the default base indicator */

unsigned char default_base_indicator (int const argument_count, char const * const argument_array [])

/* Returns the value corresponding to the position of the character in the character array */

unsigned char value (unsigned char position)

/* Converts a string representing an integer into an integer */

unsigned long int string_as_integer (char const * const string, unsigned char const base_indicator, unsigned char * const error)

int main (int const argument_count, char const * const argument_array [])