Self-Square Primes: primes whose square is a sum of consecutive primes starting at themselves
Henry John Wynn
superintelligentia.ai
First published 26 September 2026 at https://superintelligentia.ai/notes/self-square-primes/

DEFINITION
A prime p is a self-square prime if p^2 equals the sum of consecutive primes beginning at p:
    p + p' + p'' + ... + q = p^2
where p, p', p'', ..., q are consecutive primes.

RESULT
Below 150,000,000 there are exactly three self-square primes:

    p        p^2           terms   last prime added
    13       169           7       37
    463      214,369       191     1,811
    10,301   106,110,601   3,691   48,091

    13 + 17 + 19 + 23 + 29 + 31 + 37 = 169 = 13^2
    463 + 467 + ... + 1811 = 214,369 = 463^2
    10301 + 10303 + ... + 48091 = 106,110,601 = 10301^2

Companion sequences: number of terms 7, 191, 3691; last prime added 37, 1811, 48091.

NOTES
1. For a fixed number of terms k, the sum grows like kp and p^2 grows faster, so only finitely many p can work for each k; seven is the smallest k for which any prime works, and 13 is the only prime that works for k = 7.
2. Every solution uses an odd number of terms (parity of p^2 for odd p).
3. Heuristic: the sum of primes from p to q is about (q^2 - p^2)/(2 ln q); the chance that a partial sum lands exactly on p^2 is about 1/q(p) with q ~ p*sqrt(2 ln p); the sum over primes of 1/q(p) converges. Expected count below 1.5*10^8 is about 0.8; expected count in the entire tail beyond is about 0.3. A fourth self-square prime probably does not exist, and if it does it is far beyond home-computer range.
4. Search method: bit-packed sieve to 1.1*10^9, two-pointer running sum over all starting primes below 1.5*10^8, linear in the number of primes. Independent Python check to 50,000 reproduces all three.
5. Prior art found: 169 = 13 + ... + 37 is recorded (Wikipedia "169 (number)"; OEIS A132956, n = 7; OEIS A163244 lists squares that are sums of consecutive primes). No OEIS entry beginning 13, 463, 10301 and no Prime Curios! entry for the self-referential property was found as of the publication date. This is the result of a search, not a proof of novelty; earlier references are welcome at hello@superintelligentia.ai.

Copyright 2026 Henry John Wynn. Free to quote and reproduce with attribution.