Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.
Updated
Earlier bounds and status reports
cutoff_10: Fully numerical Padé bounds and completed finite certificates settle indices 12 and 14, lowering the complete cutoff to 10.
cutoff_15: The preceding cutoff-15 proof and effective small-index reductions, followed by the complete cutoff 10.
cutoff_21: The preceding cutoff-21 proof and eight complete index certificates, retained as inputs to the subsequent cutoff 15.
effective_index_24: Two certified applications of Bennett–Filaseta–Trifonov's Padé theorem yield an effective S-part exponent saving 1.3237, exceeding the 1.32 needed at i=24.
effectivity: Every i>=9 and i=1,2,7 is excluded; four indices at least four still lack effective bounds.
further_effective_bounds: The preceding seven numerical stopping bounds, subsequently closed by complete interval proofs; sharper algebraic constants are retained.
index_12_effective: Two additional asymmetric prime-pair estimates and a finite family of position-dependent interpolation certificates give an effective stopping bound at index twelve.
index_7_numerical_tail: Five numerical pair sources, 258 joint inequalities and exact relaxed polynomial coverage give n<10^3906174477003 for every bad triple at seven.
index_9_numerical_tail: Explicit joint Padé constants, weaker initial pair endpoints, and exact polynomial coverage give n<10^11514327437548 for every bad triple at index nine.
interpolation_extensions: Monotone weighted interpolation, asymmetric prime-pair bounds, and exact modular jet ranks make both indices 15 and 20 effective.
seven_nine_bounds: Both seven and nine are completely excluded. The complete cutoff is nine; four fixed indices at least four still lack effective bounds.
small_index_effective: The earlier prime-pair and interval proof of cutoff 26, retained as an input to the current cutoff 21 and complete small-index exclusions.
These notes preserve earlier stages of the calculation. For the current classification, use the research guide. Their numerical arguments remain available, but their lists of unresolved indices should be read in their historical context.