{"schema_version":"dataset-v1","id":"euclid-primes","version":1,"title":"Euclid’s argument, computed","description":"For n = 1, …, 12: the number n! + 1 and its smallest prime factor p. Every prime at most n divides n!, so it leaves remainder 1 on n! + 1; hence p is larger than n.","license":"CC0-1.0 (exact values)","problems":["prime-infinitude"],"statements":[{"problem":"prime-infinitude","statement":"prime-strictly-above","role":"Each row gives a prime p strictly above n."},{"problem":"prime-infinitude","statement":"prime-divisor-characterization","role":"Primality of p: its only divisors are 1 and p."}],"figure":{"family":"exact-table","title":"Euclid’s argument, computed","caption":"Each n! + 1 has a smallest prime factor larger than n, exactly as the proof predicts; sometimes n! + 1 is itself prime.","columns":["n","n! + 1","smallest prime factor p","as p × cofactor"],"boundary":"Twelve instances of the argument, computed exactly. The theorem for every n is the linked statement and its formal proof."},"provenance":{"method":"Exact integers; smallest prime factor by trial division; each row checks that no number from 2 to n divides n! + 1.","software":{"python":"3.12.15"},"precision":"Exact integer arithmetic.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[],"marks":[],"rows":[{"key":"n_1","cells":["1","2","2","prime"]},{"key":"n_2","cells":["2","3","3","prime"]},{"key":"n_3","cells":["3","7","7","prime"]},{"key":"n_4","cells":["4","25","5","5 × 5"]},{"key":"n_5","cells":["5","121","11","11 × 11"]},{"key":"n_6","cells":["6","721","7","7 × 103"]},{"key":"n_7","cells":["7","5,041","71","71 × 71"]},{"key":"n_8","cells":["8","40,321","61","61 × 661"]},{"key":"n_9","cells":["9","362,881","19","19 × 19,099"]},{"key":"n_10","cells":["10","3,628,801","11","11 × 329,891"]},{"key":"n_11","cells":["11","39,916,801","39,916,801","prime"]},{"key":"n_12","cells":["12","479,001,601","13","13 × 36,846,277"]}]},"sha256":"704665824e8597a62131b7bf011d2423260dd1418a696878ebe9712dbd17561c"}