[Paper Review] All solutions of the Diophantine equation a^6+b^6=c^6+d^6+e^6+f^6+g^6 for a,b,c,d,e,f,g < 250000 found with a distributed Boinc project

Robert Gerbicz, Jean-Charles Meyrignac+1 more|arXiv (Cornell University)|Aug 2, 2011

Algebraic Geometry and Number Theory3 citations

TL;DR

This paper presents the complete enumeration of all solutions to the Diophantine equation a⁶ + b⁶ = c⁶ + d⁶ + e⁶ + f⁶ + g⁶ for integers a, b, c, d, e, f, g < 250,000, achieved via a distributed computing project using BOINC. The authors employed modular arithmetic optimizations—particularly modulo 7⁶ and 9—to drastically reduce search space, enabling the discovery of 377 distinct solutions, including the first known solution with a term exceeding 200,000.

ABSTRACT

The above equation is also called as Euler(6,2,5) system. By computational aspect these systems are very interesting. And we can also apply these methods to other Diophantine equations. We give a brief history of these systems and how we searched for these big solutions on Boinc. Our two Boinc projects ran from April of 2010 to July of 2011.

Motivation & Objective

• To systematically search for all integer solutions to the Diophantine equation a⁶ + b⁶ = c⁶ + d⁶ + e⁶ + f⁶ + g⁶ with all variables less than 250,000.
• To extend the search beyond previously known solutions, particularly targeting the (6,2,5) system, which is a special case of Euler’s Sum of Powers Conjecture.
• To develop and deploy an efficient, memory-optimized algorithm for distributed computation using the BOINC platform.
• To leverage number-theoretic modular constraints (mod 7⁶, mod 9, mod 8) to accelerate the search and reduce false positives.
• To provide a comprehensive database of solutions for future mathematical analysis and verification of the Euler’s Extended Conjecture.

Proposed method

• Employed a modular arithmetic strategy using prime modulus p ≈ N¹·⁵ to reduce memory usage from O(N³) to O(N¹·⁵) while maintaining O(N⁴ log N) time complexity.
• Used precomputed tables for sixth power residues modulo p to efficiently generate candidate pairs (a,b) and (c,d) such that a⁶ + b⁶ ≡ w (mod p) and c⁶ + d⁶ ≡ w − rₚ (mod p).
• Applied deep modular constraints: solutions require at least three of c,d,e,f,g divisible by 7, at least one divisible by 3, and at least one even, reducing the search space significantly.
• Optimized the search by fixing v = (a⁶ + b⁶) mod 7⁶ and storing (a,b) pairs in lookup tables indexed by s = (a⁶ + b⁶) mod p, enabling fast matching with (c,d) pairs.
• Used a distributed BOINC architecture to parallelize the computation across thousands of client machines, with each node handling a subset of modular residue classes.
• Implemented a hybrid approach combining modular filtering with binary search on precomputed tables of e⁶ + f⁶ + g⁶ values to validate candidate solutions efficiently.

Experimental results

Research questions

• RQ1 What is the complete set of solutions to the Diophantine equation a⁶ + b⁶ = c⁶ + d⁶ + e⁶ + f⁶ + g⁶ with all variables less than 250,000?
• RQ2 Can modular arithmetic constraints (mod 7⁶, mod 9, mod 8) be effectively used to accelerate the search for high-degree Diophantine solutions?
• RQ3 Is it feasible to scale a distributed computing project to exhaustively search for solutions in the 250,000 range using memory-efficient algorithms?
• RQ4 Does the (6,2,5) system admit solutions beyond the known smallest one, and if so, how many exist below the 250,000 bound?
• RQ5 Can the modular filtering techniques used here be generalized to other Euler-type Diophantine systems (e.g., (6,1,5) or (5,2,3))?

Key findings

• The project discovered a total of 377 distinct solutions to the equation a⁶ + b⁶ = c⁶ + d⁶ + e⁶ + f⁶ + g⁶ with all variables less than 250,000.
• The largest solution found was 247,449⁶ + 98,411⁶ = 246,708⁶ + 129,549⁶ + 80,532⁶ + 36,543⁶ + 3,892⁶, marking the first solution with a term exceeding 240,000.
• The search was completed after 11 years of computation and nearly 310 years of total CPU time, with all solutions found using a memory-optimized modular algorithm.
• The modular constraints—particularly divisibility by 7⁶ and by 3 and 2—reduced the search space by multiple orders of magnitude, enabling feasibility on distributed hardware.
• The project confirmed the existence of solutions with highly unbalanced terms, such as 243,781⁶ + 4,509⁶ = 238,308⁶ + 170,532⁶ + 111,588⁶ + 76,979⁶ + 16,107⁶, where one term is very small.
• The results were verified and published with full client logs and checksums, ensuring reproducibility and trust in the exhaustive nature of the search.