Problem 977

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claims/: The 2 claim pages of Problem 977, one per claimant's result; the problem's standing derives from them.

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Statement. If $P(m)$ is the greatest prime divisor of $m$, then is it true that

$$\frac{P(2^n-1)}{n}\to \infty$$

as $n\to \infty$?

Status. Proved. The settling result is recorded on the claim page Stewart 2013.

Source. erdosproblems.com/977, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #977, https://www.erdosproblems.com/977.

References.

• [Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
• [La21] L. Lai, On the largest prime divisor of $n!+1$. arXiv:2103.14894 (2021).
• [MuWo02] Murty, Ram and Wong, Siman, The $ABC$ conjecture and prime divisors of the Lucas and Lehmer sequences. (2002), 43-54.
• [Sc62] Schinzel, A., On primitive prime factors of $a\sp{n}-b\sp{n}$. Proc. Cambridge Philos. Soc. (1962), 555-562.
• [St13] Stewart, Cameron L., On divisors of Lucas and Lehmer numbers. Acta Math. (2013), 291-314.
• [St74b] Stewart, C. L., The greatest prime factor of $a\sp{n}-b\sp{n}$. Acta Arith. (1974/75), 427-433.

Formalization. Statement in formal-conjectures, added on 2026-09-20 and pinned at that commit. It tags erdos_977 as research solved with answer(True) and carries a formal_proof pointer to Erdos977.lean in Boris Alexeev's repository https://github.com/plby/lean-proofs; its variants for Schinzel's bound, Stewart's 2013 bound and Lai's factorial bound are tagged research solved and stated with sorry, and the factorial variant $P(n!+1)/n\to\infty$ is tagged research open. The Lean proof was not built or audited here, so the catalog's label supplies no formal-verification credit; the claim page gives the pinned link.

Current assessment

Stewart's published 2013 result proves the stated full limit; its claim page Stewart 2013 records the acceptance evidence.

Status searches (UTC) covered the arXiv record, Stewart's refereed-publications list, the author-hosted published paper, exact-title correction searches, and indexed announcements including X; they found no correction or retraction of the theorem. The published theorem, rather than a negative search alone, supports the proved status.

Proof coverage. The account of Stewart's 1975 paper rests on printed pp. 427--428 for its statements and applications, and on printed p. 429 for the preliminaries and the beginning of the proof of Theorem 1. The full 1975 proofs on printed pp. 429--432 have not been reconstructed or independently verified here. The statements, hypotheses, version locators, and application scopes of Stewart's published 2013 paper and Lai's preprint are taken from the papers themselves. The specialization $a=2,b=1$ and the resulting elementary limit are checked below. Stewart's 2013 analytic proof and its same-paper lemmas, and Lai's theorem proof, have not been reconstructed or independently verified here. There is no source-access gap for the status-defining theorem; its proof is unreviewed here.

Progress

The direct integer specialization, equation (1.8), says that for fixed integers $a>b>0$,

$$P(a^n-b^n)>n\exp\!\left(\frac{\log n}{104\log\log n}\right)$$

for every sufficiently large $n$, with the threshold depending on the number of distinct prime factors of $ab$. This is on printed p. 294 of the published Acta Mathematica paper; its threshold clause continues on printed p. 295. The source result page records equation (1.8) alongside Theorem 1.1.

Taking $a=2$ and $b=1$ gives

$$\frac{P(2^n-1)}{n}> \exp\!\left(\frac{\log n}{104\log\log n}\right)\longrightarrow\infty,$$

because $\log n/\log\log n\to\infty$. The bound holds eventually for every integer $n$, so it proves the full limit, not only a statement along a subsequence. The value at $n=1$, where $2^n-1=1$, is irrelevant to this limit; Stewart uses the convention $P(1)=1$.

Known Results

Historical partial progress. Stewart's 1975 paper ([St74b], 1974/75) uses Theorem 1, printed p. 427, and the transfer following equation (4) on printed p. 428 to show, for fixed relatively prime integers $a>b>0$ and each fixed $0<\kappa<1/\log2$, that $P(a^n-b^n)/n\to\infty$ as $n\to\infty$ through integers $n>2$ with at most $\kappa\log\log n$ distinct prime factors. Stewart states on printed pp. 427--428 that a covered set has natural density one and contains every sufficiently large prime; our explicit choice of any fixed $1<\kappa<1/\log2$ clarifies the parameter range in his almost-all remark. His Theorem 2, printed p. 428, also gives effective lower bounds for the greatest prime factors of $\Phi_p(a,b)$ and $\Phi_{2p}(a,b)$ for all sufficiently large primes $p$. With $a=2,b=1$, these are restricted-exponent predecessors to the full limit proved in 2013; the restricted limit is recorded on its claim page.

Schinzel [Sc62] proved $P(2^n-1)>2n$ for $n>12$; Stewart's introduction describes the result as $P(a^n-b^n)\ge 2n+1$ for coprime $a,b$ with $ab$ a square or twice a square, outside $n=4,6,12$ when $(a,b)=(2,1)$, obtained from two primitive prime divisors. A bound with bounded ratio, it settles no instance of the limit, so it has no claim page.

The source theorem. Stewart's Theorem 1.1 gives the corresponding lower bound for Lucas--Lehmer cyclotomic factors under its stated hypotheses. Equation (1.8) is the paper's own integer specialization and is the result used above. The 18-page arXiv:1008.1274v1 manuscript is a different edition; the published equation numbers and page locators above refer specifically to the Acta Mathematica edition.

The factorial variant is separate. Lai's Theorem 1.1 in arXiv:2103.14894v1, physical pp. 1--2, gives for every nonzero polynomial $f\in\mathbb Z[X]$

$$\limsup_{n\to\infty}\frac{P(n!+f(n))}{n}\geq1+9\log2.$$

For every $\varepsilon>0$, the inputs satisfying $n!+f(n)>1$ and $P(n!+f(n))>(1+9\log2-\varepsilon)n$ have positive lower density. The choice $f=1$ concerns $n!+1$, the related question mentioned in the catalog remarks. It is a different sequence and a different conclusion from $P(2^n-1)/n\to\infty$; this finite limsup lower bound does not prove a divergent factorial limit either.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• lai_2021_largest_prime_divisor
• lai_2021_largest_prime_divisor / least_prime_divisor_bound
• lai_2021_largest_prime_divisor / lemma_2_7
• lai_2021_largest_prime_divisor / theorem_1_1
• stewart_2013_divisors_lucas_lehmer
• stewart_2013_divisors_lucas_lehmer / lemma_4_3
• stewart_2013_divisors_lucas_lehmer / theorem_1_1
• stewart_2013_divisors_lucas_lehmer / theorem_1_2
• stewart_nd_greatest_prime_factor
• stewart_nd_greatest_prime_factor / lemma_3
• stewart_nd_greatest_prime_factor / theorem_1
• stewart_nd_greatest_prime_factor / theorem_2
• erdos_1965_recent_advances_current_problems_number_theory