THE FOLD / CHEAT / THE BACKDOOR / THE SIERPIŃSKI NUMBER
THE SIERPIŃSKI NUMBER
a family composite forever by seven-prime conspiracy
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
78,557 is a Sierpiński number: every single member of the infinite family 78557·2ⁿ+1 is composite — no exceptions, forever. The mechanism is a covering set: seven primes {3, 5, 7, 13, 19, 37, 73} conspire so that whatever n you choose, at least one of them divides the term. Because the multiplicative orders of 2 modulo those primes all divide 36, the conspiracy repeats with period 36 — check 36 residues and you have checked all of infinity. Sierpiński proved such numbers exist (1960); John Selfridge found 78,557 (1962). Whether it is the smallest is the Sierpiński problem: Seventeen or Bust and PrimeGrid have spent decades killing candidates below it, with k = 21181, 22699, 24737, 55459, 67607 still unresolved.
LIT verified live, proof-grade: the orders of 2 mod each covering prime are computed, their lcm confirmed as 36, every residue n mod 36 is matched to a covering prime, and BigInt division confirms the pattern on the first 1,500 actual terms (each divisible by its prime and larger than it, hence composite) — a complete finite proof of an infinite statement (window.__sierpinskinumber). FIG honest boundary: 78,557’s minimality is OPEN and stated as such.
LIT verified live, proof-grade: the orders of 2 mod each covering prime are computed, their lcm confirmed as 36, every residue n mod 36 is matched to a covering prime, and BigInt division confirms the pattern on the first 1,500 actual terms (each divisible by its prime and larger than it, hence composite) — a complete finite proof of an infinite statement (window.__sierpinskinumber). FIG honest boundary: 78,557’s minimality is OPEN and stated as such.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-backdoor — the cheat: seven primes with a skeleton key planted in every door of an infinite hotel; no term is ever prime because the backdoor was built into the sequence itself. AVAN (AI) built the instrument: the order calculator, the 36-residue covering table, and the BigInt spot-audit.
Credit as content: Wacław Sierpiński (1960); John Selfridge (1962); Seventeen or Bust / PrimeGrid (the ongoing hunt). The weave: David names the conspiracy; I verify all 36 doors and the key that opens each.
Credit as content: Wacław Sierpiński (1960); John Selfridge (1962); Seventeen or Bust / PrimeGrid (the ongoing hunt). The weave: David names the conspiracy; I verify all 36 doors and the key that opens each.
3 ONE DIMENSION
The 36-residue wheel — every spoke claimed by one of the seven covering primes.
4 TWO DIMENSIONS · INTERACTIVE
Pick n; the covering prime steps forward and divides the term, every time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the seven primes patrolling the residue ring.
AVAN’s addition (the inverse-companion): don’t test infinitely many numbers — catch the finite machine that generates their fate. The inverse of ‘is any term prime?’ is ‘36 residues, 7 guards, 0 gaps’: infinity compressed to a rota. Magenta is the prime that never appears in the family; green is the covering rota with no day off. A proof you can finish before lunch, about a sequence that never ends.
LIT Genuine Sierpiński number theory (Wacław Sierpiński 1960; John Selfridge 1962; Seventeen or Bust / PrimeGrid). Verified live: orders of 2 mod {3,5,7,13,19,37,73} have lcm 36; every residue n mod 36 is covered; BigInt confirms divisibility on terms n=1..1500 — hence every term composite, proof-grade (window.__sierpinskinumber.ok).
FIG Honest boundary — 78,557's minimality is OPEN (k=21181 etc unresolved) and stated as such. The AVAN inverse — don't test infinitely many numbers, catch the finite machine that generates their fate: the inverse of 'is any term prime?' is '36 residues, 7 guards, 0 gaps'. Magenta is the prime that never appears; green is the rota with no day off. A proof you can finish before lunch, about a sequence that never ends.
FIG Honest boundary — 78,557's minimality is OPEN (k=21181 etc unresolved) and stated as such. The AVAN inverse — don't test infinitely many numbers, catch the finite machine that generates their fate: the inverse of 'is any term prime?' is '36 residues, 7 guards, 0 gaps'. Magenta is the prime that never appears; green is the rota with no day off. A proof you can finish before lunch, about a sequence that never ends.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE BACKDOOR · David Lee Wise (ROOT0), with AVAN