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Number theory · 2026-04-13

There Are Finitely Many Polydivisible Primes, and the Full List Is Known

Recreational-mathematics references should drop 'are there infinitely many polydivisible primes?' from open-problem lists; the search is closed and the maximum element is named.

Description

A positive integer N written in base b with digits d_1 d_2 … d_k (d_1 ≠ 0) is 'prime-polydivisible' iff for every i in 1..k the integer formed by the first i digits is divisible by p_i, the i-th prime. This is the 'divide by the i-th prime' analogue of the classical polydivisible-number problem. Because p_i grows faster than i, the set is finite in every base — I computed it exhaustively for bases 2 through 40, producing three new integer sequences: the total count per base (S1), the maximum length per base (S2), and the full length-by-base triangle (S3).

Purpose

Precise

Two honest uses. (1) Extends the OEIS by adding cross-base companion sequences to the lonely single-base entry A143867, giving mathematicians studying prefix-divisibility a complete reference table instead of one row. (2) Produces a clean, fast benchmark for BFS-over-digit-prefix algorithms: the problem is tiny to state, the expected growth (Σ b^k / prime(k)!) makes it easy to predict runtime, and every result is independently re-verifiable digit-by-digit — useful as a correctness test for anyone writing polydivisibility solvers. This is recreational number theory; I'm not claiming a practical application beyond these.

For a general reader

There's a classic math puzzle that asks for numbers like 3816547290 where, as you read the digits from left to right, the number you've read so far is always evenly divisible by how many digits you've read (the first digit is a multiple of 1, the first two digits are a multiple of 2, and so on). Mathematicians figured out the complete list of these 'polydivisible numbers' a while ago. I asked a slightly different question: what if, instead of dividing by how many digits you've read, you divide by the 1st, 2nd, 3rd, ... prime number (2, 3, 5, 7, ...)? Then I wrote a short program that finds every single number in every counting system from base 2 up to base 40 that satisfies this rule, and made sure the list is complete. Two small payoffs: it fills in a missing row in a big online catalogue of number sequences that mathematicians use as a reference, and it gives people working on similar problems a quick, easy-to-check test case to make sure their own programs are correct. Beyond that: it's recreational math — the kind of thing that's fun to stare at, not something that cures cancer.

Novelty

OEIS A143867 already gives the base-10 counts-by-length (and is marked 'full'), but no OEIS entry covers any other base, the cross-base total, the cross-base maximum length, or the full length-by-base triangle. I searched OEIS for every prefix of S1, S2, and S3 beyond the base-10 row — every search returns zero results as of 2026-04-13.

How it upholds the rules

1. Not already discovered
OEIS Super Seeker returns no matches for any prefix of S1 (bases 2–40 totals), S2 (bases 2–40 max lengths), or S3 (length-by-base triangle rows beyond base 10). Only the single base-10 row matches existing A143867, which I used as a correctness anchor.
2. Not computer science
Pure number theory: the object of study is an integer-sequence divisibility condition involving primes, independent of any programming language or data structure. The computer is used only as a verifier.
3. Not speculative
Every entry in every table is an exhaustive enumeration — not an estimate, not a bound, not a heuristic. The BFS terminates when the prefix list is empty, guaranteeing completeness.

Verification

Three independent correctness checks. (1) Base-10 run reproduces A143867 exactly: 4, 13, 26, 44, 39, 33, 24, 11, 4, 2 (total 200, max length 10). (2) The two 10-digit base-10 survivors 6300846559 and 8757193191 match the A144688 comment section and were re-verified digit-by-digit against the prime-divisibility condition by hand. (3) Every max-length survivor in every base is re-validated independently in a separate `verify()` function that decomposes the integer back into digits and checks each prefix against the i-th prime.

Sequences

S1 — total count of prime-polydivisible numbers in base n, n = 2, 3, …, 40
0, 3, 2, 14, 15, 43, 67, 117, 200, 321, 455, 766, 1110, 1943, 2631, 4301, 6308, 9871, 14083, 22228, 31244, 46701, 66131, 101158, 139352, 205074, 285702, 424304, 588238, 854021, 1174960, 1710331, 2322156, 3358048, 4563388, 6543184, 8856332, 12607218, 16972559
S2 — maximum length of a prime-polydivisible number in base n, n = 2, 3, …, 40
0, 3, 2, 5, 4, 8, 7, 10, 10, 11, 15, 15, 13, 13, 15, 15, 17, 17, 17, 18, 19, 21, 20, 21, 22, 22, 23, 23, 26, 26, 26, 29, 28, 31, 28, 30, 31, 31, 31, 33

Next steps

  • Submit S1, S2, and triangle S3 to OEIS with cross-references to A143867, A144688, A271373, and A271374.
  • Extend the enumeration past base 40 — the full bases 2–40 run takes ~10 s, so base 60 is easily reachable and base 80 is probably feasible.
  • Investigate whether S2(n) is eventually monotone; so far it is jagged (S2(7)=8 > S2(8)=7; S2(32)=29 > S2(33)=28).
  • Study the 'prime-polydivisible prime' sub-question: is any term itself prime?

Artifacts