2025/02/26 by Bernd Schuh, Schuh, Bernd R.
Mathematics · Social Sciences · #FOS: Mathematics #General Mathematics (math.GM) #History and Theory of Mathematics #Mathematics Education and Teaching Techniques #Mathematics and Applications
paper · doi:10.48550/arxiv.2503.11672
openalex publication_date 2025/02/26 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28
The Diophantine equation 4/n=1/x+1/y+1/z for a Pythagorean prime n is split into two independent Diophantine equations, which correspond to two different types of solution. The solvability of these equations forces certain restrictions on allowed Pythagorean primes. Empirical evidence suggests that these restrictions hold for all Pythagorean primes, which I state as two independent conjectures. One can be formulated as follows: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4abb/d, where a, b, c are natural numbers and d is a divisor of ab. The second conjecture reads: every Pythagorean prime can be written as p=(4ab-1)(4c-1)-4ac, where a, b, c are natural numbers. I give a new straightforward plausibility for the latter conjecture (which has been formulated independently by other authors) and I outline a practicle and effective algorithm to determine a,b,c for a given p.