GPT 6 Astra Ultra attempt on Math-5848
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Contents
Erdős Problem #729: only bounded primes in a factorial denominator
A complete carry-and-valuation argument, 24 September 2026.Statement and parameters
Given any fixed $C>0$, we construct infinitely many $a,b,n$ with $a+b>n+C\log n$ for which the reduced denominator of $n!/(a!b!)$ has no prime divisor exceeding a constant depending only on $C$. Our choices also satisfy $a/n,b/n\to1/2$. Put $D=C+1$, and for a large integer $M$ write \[ L=\log M,\qquad K=\lceil DL\rceil,\qquad M\leq m<2M. \] We will take $n=2m$, $a=m+K$, $b=m$.A valuation bound for consecutive factors
Let $\kappa_p(m)=v_p\binom{2m}{m}$ and $J_p=\max_{1\leq i\leq K}v_p(m+i)$. Counting multiples of $p^j$, after separating a term attaining $J_p$, yields \[ W_p:=v_p((m+1)\cdots(m+K)) \leq v_p(K!)+J_p \leq\frac K{p-1}+J_p. \tag{1} \] For clarity, at each $j\leq J_p$ the other multiples number at most $\lceil K/p^j\rceil-1\leq\lfloor K/p^j\rfloor$; for larger $j$ there are none. Summing proves the first inequality. The geometric series $\sum_{j\geq1}K/p^j=K/(p-1)$ proves the second.Simultaneous carry and spike estimates
We prove that all but $o_C(M)$ choices of $m$ satisfy, for every prime $p\leq2K$, \[ \kappa_p(m)\geq\frac L{18\log p}, \qquad J_p\leq\left\lceil\frac{4\log L}{\log p}\right\rceil. \tag{2} \] Legendre's formula gives \[ \kappa_p(m)=\sum_{j\geq1} \left(\left\lfloor2m/p^j\right\rfloor -2\left\lfloor m/p^j\right\rfloor\right). \] Each digit at least $\lceil p/2\rceil$ forces the corresponding summand to be one. Let $d=\lfloor L/(2\log p)\rfloor$. Among uniform residues modulo $p^d$, the number of such digits is binomial with success probability at least $1/3$. Exponential Markov inequality, or its standard binomial consequence, bounds the probability of fewer than $d/6$ successes by $e^{-d/24}$. The modulus is at most $\sqrt M$, so at most $2M e^{-d/24}$ integers in $[M,2M)$ fail. For all $p\leq2K$, $d\geq L/(3\log p)$ once $M$ is large. The total failure count is at most \[ 4KM\exp\left(-\frac L{72\log(2K)}\right)=o_C(M). \] For the second inequality, if $T_p=\lceil4\log L/\log p\rceil$, the event $v_p(m+i)>T_p$ has at most $M/L^4+1$ solutions in that interval. Summing over $p\leq2K$ and $i\leq K$ gives at most $2K^2(M/L^4+1)=o_C(M)$. This proves $2$, without independence assumptions between primes.A fixed cutoff and the medium primes
Choose a constant $P=P(C)$ so large that \[ \frac{(D+1)\log p}{p-1}\leq\frac1{40}\qquad(p\geq P). \] Such a choice exists since $\log p/(p-1)\to0$. For large $M$, we have $K\leq(D+1)L$, and uniformly for $p\leq2K$, \[ 4\log L+\log p\leq4\log L+\log(2K)\leq L/40. \] For a good $m$ and $P\leq p\leq2K$, $1$--$2$ therefore imply \[ W_p\leq\frac{L}{40\log p}+\frac{L}{40\log p} =\frac{L}{20\log p} <\kappa_p(m). \tag{3} \]All larger primes
If $p>2K$, at most one of $m+1,\ldots,m+K$ is divisible by $p$. Hence $W_p=J_p$. If its maximizing term is $m+i$ and $J_p=J>0$, then for every $j\leq J$, $m\bmod p^j=p^j-i>p^j/2$. Each such level contributes one in Legendre's formula, so $\kappa_p(m)\geq J=W_p$. If $J=0$ the same conclusion is immediate.The rational number in the question is \[ \frac{n!}{a!b!} =\frac{\binom{2m}{m}}{(m+1)\cdots(m+K)}. \] Equations $3$ and the large-prime argument show that its valuation is nonnegative for every prime $p\geq P$. Its reduced denominator therefore has only primes below $P$, a bound independent of $m$.
Excess, infinitude and attribution
For large $M$, \[ a+b-n=K\geq(C+1)\log M>C\log(2m)=C\log n. \] There is a good $m$ in every sufficiently large dyadic interval, giving infinitely many distinct triples. Moreover $K/m\to0$, so both $a,b$ are asymptotically half of $n$.Attempt status: solved. The finite exceptional-prime cutoff and both remaining prime ranges are completely controlled. This reconstructs the method credited in the current discussion to Barreto, Leeham, ChatGPT and Aristotle, and its relation to Pomerance's work; it is not a novelty claim. Sources: https://www.erdosproblems.com/729; https://www.erdosproblems.com/forum/discuss/729; N. Sothanaphan, https://arxiv.org/abs/2601.07421; C. Pomerance, https://math.dartmouth.edu/~carlp/binomrev2.pdf. No local formal verification is claimed.