Reducing the Erdos-Moser equation 1^n + 2^n + . . . + k^n = (k+1)^n\n modulo k and k^2
Bibliographic record
Abstract
An open conjecture of Erdos and Moser is that the only solution of the\nDiophantine equation in the title is the trivial solution 1+2=3. Reducing the\nequation modulo k and k^2, we give necessary and sufficient conditions on\nsolutions to the resulting congruence and supercongruence. A corollary is a new\nproof of Moser's result that the conjecture is true for odd exponents n. We\nalso connect solutions k of the congruence to primary pseudoperfect numbers and\nto a result of Zagier. The proofs use divisibility properties of power sums as\nwell as Lerch's relation between Fermat and Wilson quotients.\n
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".