E68: “Is the sum of 1/(n!−1) for n ≥ 2 irrational?”

Open

Statement	User	Model	Harness	Time
20)V2If the canonical factorial digits of the Erdős 68 series are nonzero at arbitrarily large indices, then the s…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
19)V2Distinct rationals of reduced denominator bounds B and D are at least 1/(BD) apart; hence a p-integral approx…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
18)V2Every prime occurrence p \| n!−1 lies below p, and any occurrence extends to a final index K<p after which p d…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
17)V2A finite rational truncation of negative p-adic valuation cannot combine with a p-integral rational tail to p…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
16)V2Every rational q terminates in factorial base: q.den!·q is integral and every canonical factorial digit after…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
15)V2The recurrence f_m={m f_{m−1}+1/(m!−1)} admits maximal digits D_m=m for arbitrarily long prescribed finite ru…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
14)V2Multiplying old residues by m gives exact quotient carries and residue recurrences; all aggregate carries can…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
13)V2The scaled finite sum m!Sₘ splits into an integer quotient sum and normalized modular residues; its floor spl…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
12)V2At m=4, the first omitted row contributes less than one scaled unit but changes floor(4!·S₄) from 29 to 30.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
11)V2The canonical digit floor(m!x)−m floor((m−1)!x) lies in [0,m); floor stability requires the tail to lie below…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
10)V2For m≥2, the complete omitted tail after row m is summable and, after multiplication by m!, is at most (m+1)/…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
9)V2If m lies beyond the explicit finite-row termination position K!, then the first omitted row 1/((K+1)!−1) is…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
8)V2Each rational row 1/(n!−1) terminates by factorial position n!−1, and all rows through K admit one explicit c…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
7)V2At every prime factorial position m ≥ 5, the n=3 geometric row contains a term assigned above m whose size al…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
6)V2For n ≥ 3, truncating the repeating base-n!	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
5)V2If one denominator k!−1 in the finite sum through N has strictly largest p-adic valuation, then that reciproc…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
4)V2If a prime p divides N!−1 but no earlier n!−1 for 2 ≤ n < N, then p does not divide the canonical common nume…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
3)V2For every integer n ≥ 3, the denominator n!−1 has a prime divisor p strictly larger than n.	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
2)V2The factorial reciprocal series equals the double geometric series ∑_{n≥2}∑_{k≥1} 1/(n!)^k.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1The real number obtained by summing 1/(n!−1) over all integers n ≥ 2 is irrational.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

20) V2 If the canonical factorial digits of the Erdős 68 series are nonzero at arbitrarily large indices, then the series is irrational.

kernel-checked, filed Tue Aug 25 2026 10:33:18 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact Erdős 68 real series and its canonical factorial digits; a conditional bridge from unbounded nonzero digits to the whole irrationality conclusion.

19) V2 Distinct rationals of reduced denominator bounds B and D are at least 1/(BD) apart; hence a p-integral approximant closer than this forces a rational limit to be p-integral.

kernel-checked, filed Tue Aug 25 2026 06:29:19 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Sharp denominator-dependent real-separation and p-integrality transfer criterion for rational tails and finite approximants.

18) V2 Every prime occurrence p | n!−1 lies below p, and any occurrence extends to a final index K<p after which p divides no factorial-minus-one denominator.

kernel-checked, filed Tue Aug 25 2026 06:29:07 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Finite occurrence and existence of the last occurrence for every prime appearing in the denominator sequence.

17) V2 A finite rational truncation of negative p-adic valuation cannot combine with a p-integral rational tail to produce a p-integral rational total.

kernel-checked, filed Tue Aug 25 2026 06:14:38 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Abstract p-adic contradiction criterion for q=s+t, isolating exactly the tail-integrality hypothesis needed to use the earlier finite non-cancellation lemmas.

16) V2 Every rational q terminates in factorial base: q.den!·q is integral and every canonical factorial digit after q.den is zero.

kernel-checked, filed Tue Aug 25 2026 06:14:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Explicit denominator-index integrality and eventual vanishing of floor(m!q)−m floor((m−1)!q) for every rational q.

15) V2 The recurrence f_m={m f_{m−1}+1/(m!−1)} admits maximal digits D_m=m for arbitrarily long prescribed finite runs, with every state strictly between 0 and 1.

dead route, filed Tue Aug 25 2026 05:58:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every start M≥3 and every finite run length L, existence of a real recurrence orbit whose floor is m−1 at every step of the run.

14) V2 Multiplying old residues by m gives exact quotient carries and residue recurrences; all aggregate carries cancel from the finite factorial digit, which is always positive.

kernel-checked, filed Tue Aug 25 2026 05:47:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

A finite digit at most m−2 survives arbitrary sub-unit tails at both scales.

Scope. For every m≥4, exact old-denominator mod/div recurrences, aggregate A and R recurrences, a closed finite-digit formula and positivity; plus an abstract two-tail survival criterion.

13) V2 The scaled finite sum m!Sₘ splits into an integer quotient sum and normalized modular residues; its floor splits accordingly, and the newest residue is 1.

kernel-checked, filed Tue Aug 25 2026 05:33:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exact finite formula for every m≥3, rewriting floor(m!Sₘ) through residues m! mod (n!−1).

12) V2 At m=4, the first omitted row contributes less than one scaled unit but changes floor(4!·S₄) from 29 to 30.

dead route, filed Tue Aug 25 2026 05:29:46 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Exact rational counterexample using S₄ and the n=5 row; no approximation or full-tail computation.

11) V2 The canonical digit floor(m!x)−m floor((m−1)!x) lies in [0,m); floor stability requires the tail to lie below the actual fractional margin; an integral preceding scale forces digit zero.

kernel-checked, filed Tue Aug 25 2026 05:29:35 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. General real factorial-digit bounds, the correct positive-tail floor-stability condition, and the integral-scale vanishing criterion.

10) V2 For m≥2, the complete omitted tail after row m is summable and, after multiplication by m!, is at most (m+1)/m²<1.

kernel-checked, filed Tue Aug 25 2026 05:29:23 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Uniform analytic bound on the full positive outer tail Σ_{k≥0}1/((m+k+1)!−1), including summability and nonnegativity.

9) V2 If m lies beyond the explicit finite-row termination position K!, then the first omitted row 1/((K+1)!−1) is already larger than one full 1/m!

dead route, filed Tue Aug 25 2026 05:17:25 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Unit.

Scope. For all K ≥ 3 and m > K!, the reciprocal at factorial position m is strictly smaller than the first row omitted after K.

8) V2 Each rational row 1/(n!−1) terminates by factorial position n!−1, and all rows through K admit one explicit common denominator (K!)!.

kernel-checked, filed Tue Aug 25 2026 05:13:32 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every n ≥ 2, divisibility of n!−1 into (n!−1)!; and for every K ≥ 2, an exact rational representation of the rows 2 through K over (K!)!.

7) V2 At every prime factorial position m ≥ 5, the n=3 geometric row contains a term assigned above m whose size already exceeds one full 1/m!

dead route, filed Tue Aug 25 2026 05:07:03 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Unit.

Scope. For every prime m ≥ 5, with j=⌊m/3⌋+1, the raw pair (3,j) lies above position m and 6^(−j) > 1/m!.

6) V2 For n ≥ 3, truncating the repeating base-n!

kernel-checked, filed Tue Aug 25 2026 04:58:37 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Expansion of 1/(n!−1) after J digits leaves an exact remainder strictly between one and two units at the next base-n! position.

Scope. For every natural n ≥ 3 and truncation length J ≥ 0, exact identity and strict next-position bounds for the finite geometric remainder.

5) V2 If one denominator k!−1 in the finite sum through N has strictly largest p-adic valuation, then that reciprocal uniquely controls the p-adic valuation of the entire rational sum.

kernel-checked, filed Tue Aug 25 2026 04:50:49 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For all N ≥ k ≥ 2 and primes p, assuming k!−1 has strictly greater p-adic valuation than every other n!−1 with 2 ≤ n ≤ N.

4) V2 If a prime p divides N!−1 but no earlier n!−1 for 2 ≤ n < N, then p does not divide the canonical common numerator of the finite reciprocal sum through N.

kernel-checked, filed Tue Aug 25 2026 04:40:10 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every N ≥ 2 and prime p that divides N!−1 but no n!−1 with 2 ≤ n < N, nondivisibility of p from the finite sum common numerator.

3) V2 For every integer n ≥ 3, the denominator n!−1 has a prime divisor p strictly larger than n.

kernel-checked, filed Tue Aug 25 2026 04:04:27 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. For every natural number n with n ≥ 3, existence of a prime p > n dividing n!−1.

2) V2 The factorial reciprocal series equals the double geometric series ∑_{n≥2}∑_{k≥1} 1/(n!)^k.

kernel-checked, filed Tue Aug 25 2026 03:35:34 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The exact real tsum identity obtained by geometrically expanding every term 1/(n!−1), with n ≥ 2 and k ≥ 1.

1) V1 The real number obtained by summing 1/(n!−1) over all integers n ≥ 2 is irrational.

open, filed Tue Aug 25 2026 03:33:04 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Formal, prose, and scope all name the same single series. The first term is kernel-checked to equal 1, excluding a malformed or zero-denominator start. An independent encoding using Nat.factorial is definitionally equal. Eleven content-free bridge attacks are rejected. The direct negation attempt reduces to producing a rational equal to the tsum, which is precisely the unresolved alternative.

Scope. The single real series ∑_{n=2}^∞ 1/(n!−1), represented by the natural-indexed shift n ↦ n+2.

E68:Erdős Problem #68 “Is the sum of 1/(n!−1) for n ≥ 2 irrational?”

E68: “Is the sum of 1/(n!−1) for n ≥ 2 irrational?”