GPT 6 Astra Ultra attempt on Math-5609

FORMAL RESTATEMENT

Erdős #369; reconstruction of the smooth-polynomial construction, 2026-09-24. For fixed $\varepsilon>0$ and $k\geq2$, show that every sufficiently large $N$ contains $k$ consecutive positive integers all of whose prime factors are at most $N^\varepsilon$.

QUICK LITERATURE AND CONTEXT CHECK

Taken literally, the question allows the fixed block $1,\ldots,k$. The substantive stronger statement is that such a block can be placed near $N$. Balog--Wooley established arbitrarily large smooth blocks; Theorem 2.1 of Bober--Fretwell--Martin--Wooley, Smooth values of polynomials, supplies a polynomial construction. The argument below reconstructs its linear-factor case and obtains a block inside $[N-N^c,N]$ for some fixed $c<1$. This is a reconstruction of known mathematics.

OBJECTIVE AND ATTACK PLAN

Choose disjoint prime sets so that several cyclotomic factorizations simultaneously have small degree. Evaluate the resulting polynomial at its last integer argument below $N$.

WORK

Choosing the exponents. Put $\eta=\min(\varepsilon/2,1/2)$. The divergence of the sum of reciprocals of the primes implies that there are disjoint finite sets $\mathcal P_1,\ldots,\mathcal P_k$ of odd primes such that, on writing \[ \gamma_j=\prod_{p\in\mathcal P_j}p,\quad D=\prod_{j=1}^k\gamma_j,\quad D_j=D/\gamma_j, \] one has $\varphi(\gamma_j)/\gamma_j<\eta$ for every $j$. Indeed, $\prod(1-1/p)\leq\exp(-\sum1/p)$, and deleting finitely many primes leaves the reciprocal sum divergent. All $\gamma_j$ are pairwise coprime. Choose $1\leq u_j<\gamma_j$ satisfying $D_j u_j\equiv1\pmod {\gamma_j}$, and put $\mu_j=D_j u_j$. Thus $\mu_j\equiv1\pmod {\gamma_j}$ and $\gamma_i\mid\mu_j$ if $i\ne j$. Define \[ C=\prod_{j=1}^k j^{\mu_j},\qquad g(t)=Ct^D. \] For each $j$ the positive integer \[ A_j=j^{(\mu_j-1)/\gamma_j}\prod_{i\ne j}i^{\mu_i/\gamma_j} \] is well defined and satisfies $C=jA_j^{\gamma_j}$. Consequently \[ g(t)-j=j\big((A_jt^{D_j})^{\gamma_j}-1\big) =j\prod_{e\mid\gamma_j}\Phi_e(A_jt^{D_j}), \tag{1} \] where $\Phi_e$ denotes the $e$th cyclotomic polynomial.

Bounding every prime factor. Because $\gamma_j$ is squarefree, $e\mid\gamma_j$ implies $\varphi(e)\leq\varphi(\gamma_j)$. Each polynomial factor on the right of $1$ therefore has degree at most \[ D_j\varphi(\gamma_j)<\eta D. \] There are finitely many fixed factors. For all sufficiently large integer $t$, every such factor is nonzero and has absolute value at most $K t^{\eta D}$ for one fixed $K$. Every prime dividing $g(t)-j$ divides either $j$ or one of those integer factors. Since $g(t)=Ct^D$, and $\eta<\varepsilon$, \[ P(g(t)-j)\leq\max(k,Kt^{\eta D})\leq g(t)^\varepsilon \quad (1\leq j\leq k) . \tag{2} \] No assertion about irreducibility after substitution is needed: an integer's prime divisors are bounded by its absolute value.

Positioning the block for every large $N$. Take $t=\lfloor(N/C)^{1/D}\rfloor$. Then $g(t)\leq N<g(t+1)$, and \[ 0\leq N-g(t)<C\big((t+1)^D-t^D\big)=O(N^{1-1/D}). \] The $k$ consecutive numbers $g(t)-k,\ldots,g(t)-1$ are positive for large $N$ and satisfy $2$, hence are $N^\varepsilon$-smooth. With $c=1-1/(2D)$, their distance from $N$ is eventually at most $N^c$. This proves the stronger all-$N$ statement.

RESULT AND LIMITATIONS

Attempt status: solved. The construction and the passage from polynomial values to a block near every large $N$ are proved above. The standard facts used are divergence of the prime reciprocal sum and cyclotomic factorization. No effective small threshold or optimal value of $c$ is claimed.

SOURCES

https://www.erdosproblems.com/369; Bober, Fretwell, Martin and Wooley, Smooth values of polynomials, Theorem 2.1: https://arxiv.org/html/1710.01970. Tao's explanation of the near-$N$ deduction: https://www.erdosproblems.com/forum/discuss/369.

Completion rate estimate: 100%.