Ponder This Challenge - September 2026 - Loeschian Arithmetic Progressions
| Source: IBM Research
Tags: IBM Research, number theory, computational puzzles, mathematics
IBM Research's September 2026 Ponder This puzzle asks solvers to find arithmetic progressions of at least 35 terms within Loeschian numbers (integers of the form x² + y² + xy), inspired by the Green-Tao theorem on primes and arithmetic structure in dense number sets.
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IBM Research's Ponder This series posts monthly math challenges aimed at recreational mathematicians and competitive problem-solvers. The September 2026 edition draws from the Green-Tao theorem (2004), which proved arbitrarily long arithmetic progressions exist within the prime numbers. The same structural property holds for Loeschian numbers — integers expressible as x² + y² + xy — which are dense enough to guarantee similar results.\n\nThe challenge: find a 35-term arithmetic progression from the Loeschian numbers with the smallest possible endpoint, expressed as a starting value and step size. An example in the puzzle shows that 1, 7, 13, 19 forms a 4-term progression with step 6. Bonus marks go to 42-term progressions, with an extra distinction for the longest found above 42.\n\nWhile this is a pure math puzzle with no direct AI application, IBM Research's tradition of engaging the technical community with computational challenges attracts solvers who apply combinatorial search, constraint programming, and AI-assisted search to such problems. For AI practitioners, this is recreational mathematics rather than AI news.