Luca 2001 conjecture erdos stewart

../

lemma: The elementary lemma of Luca's paper: for n at least 12 in [p_(k-1), p_k), n factorial plus one is never a power of p_k alone or of p_(k+1) alone.

theorem: Luca's proof of the Erdős–Stewart conjecture: when n lies in [p_(k-1), p_k), n factorial plus one is a product of nonnegative powers of p_k and p_(k+1) for no n at least 6.

---

F. Luca, On a conjecture of Erdős and Stewart, Math. Comp. 70 (2001), no. 234, 893--896; S 0025-5718(00)01178-9, DOI 10.1090/S0025-5718-00-01178-9. Received by the editor 4 January 1999; published electronically 8 March 2000. 2000 MSC primary 11D61.

The copy read for this card is the American Mathematical Society's publisher PDF of the four printed pages 893--896, with a text layer, identified by the DOI 10.1090/S0025-5718-00-01178-9 (https://doi.org/10.1090/S0025-5718-00-01178-9). The copy prints "©2000 American Mathematical Society" on printed p. 893, every other right reserved.

Read status: claims checked for the Theorem and the Lemma, whose statements were read clause by clause on the printed pages; their proofs were read but not verified. The statements are on the result pages linked below. The Theorem is consumed by Problem 1058's claim page, which discloses that the proof is not checked.

Contents

• The conjecture (p. 893): with $p_k$ the $k$th prime, Erdős and Stewart conjectured, as reported in Guy's Unsolved problems in number theory, Problem A2 (the paper's [3]; the problem's [Gu04] is a later edition), that every solution of

$$n!+1=p_k^{a}p_{k+1}^{b},\qquad a\ge0,\ b\ge0,\ p_{k-1}\le n<p_k \tag{1}$$

has $n\le5$.

• Theorem (p. 893): no solution of (1) has $n\ge6$. The paper notes that a direct check disposes of $5<n\le11$ and from then on assumes $n\ge12$.
• Lemma (p. 893; proof pp. 893--894): $ab\ne0$ in any solution of (1) with $n\ge12$, the paper's standing assumption from there on (each solution with $n\le5$ has $ab=0$, e.g. $4!+1=5^2$); the proof compares $\operatorname{ord}_2(n!)\ge n-\log_2(n+1)$ with the $2$-adic valuation of $p^a-1$.
• Section 3 (pp. 894--895): the Bugeaud–Laurent lower bound for $p$-adic linear forms in two logarithms (their Théorème 4, with $p=2$), applied to $\operatorname{ord}_2(n!)=\operatorname{ord}_2(p_k^ap_{k+1}^b-1)$ and combined with $p_k<p_{k+1}<2n$, gives $n<7\,242\,116$ and hence $n<p_k<p_{k+1}<7.5\cdot10^6$.
• Section 4 (p. 895): for $193<n$, a computer search over $A\in\{p_k,\ p_kp_{k+1},\ p_k^2p_{k+1}\}$ with $193<p_k<p_{k+1}<7.5\cdot10^6$ found no $A$ that is a cubic residue modulo every prime $q\le193$ with $q\equiv1\pmod3$, forcing $a\equiv b\equiv0\pmod3$ in any solution and contradicting the Erdős–Obláth theorem (Theorem EO), by which $n!$ is never $x^p\pm y^p$ with coprime $x,y$ and an odd prime $p$ (the print omits trivial solutions such as $2!=1^p+1^p$, which do not arise for $n>193$); for $n\le193$ a second computation checked $n!+1\not\equiv0\pmod{p_kp_{k+1}}$. Both computations are reported, not reproduced, in the paper.

Compiled scope

The whole four-page paper was read in the text layer and on the page images. The statements of the Theorem and the Lemma were checked clause by clause; the proofs were read but not verified (the Bugeaud–Laurent constants, the two reported computations and the Erdős–Obláth theorem were not checked). Nothing here is independently reviewed.

Bears on. #1058: the Theorem (p. 893) states that for $n\ge6$ no $n\in[p_{k-1},p_k)$ has $n!+1=p_k^ap_{k+1}^b$, so only finitely many $n$ (all $\le5$) have $n!+1$ divisible by no prime other than $p_k$ and $p_{k+1}$, a yes to the question. The Lemma (p. 893), that a solution with $n\ge12$ has $ab\ne0$, enters only through the proof of the Theorem.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.