Pasten 2024 largest prime factor improvements subexponential

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corollary_1_5: States that for an absolute kappa > 0 the largest prime factor of xy(x+y) is at least kappa (log_2 y)^2/log_3 y as x < y vary over coprime positive integers, which at x = 1 bounds the largest prime factor of n(n+1).

theorem_1_1: States that for some constant kappa > 0 the largest prime factor of n^2+1 is at least kappa (log_2 n)^2/log_3 n as n grows.

theorem_1_2: States that for some constant kappa > 0 the radical of n^2+1 is at least exp(kappa (log_2 n)^2/log_3 n) as n grows.

theorem_1_4: States that for coprime a+b=c with R=rad(abc), log c is at most eta^{-1}exp(kappa sqrt((log R)log_2 R)) when a <= c^{1-eta}, and at most q exp(kappa sqrt((log R)log_2 R)) with q the least of P(a), P(b), P(c).

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Pasten, Hector, The largest prime factor of {$n^2+1$} and improvements on subexponential {$ABC$}. Invent. Math. 236 (2024), no. 1, 373--385. https://doi.org/10.1007/s00222-024-01244-6. The copy read for this card is arXiv:2312.03566v1 (10 pages); page locators below are its pages. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2312.03566), every other right reserved.

Pasten proves (Theorem 1.1, p. 1) that the largest prime factor P(n^2+1) exceeds kappa*(log_2 n)^2/log_3 n for some constant kappa>0, nearly the square of the previously sharpest bound, which was Chowla's 1934 estimate P(n^2+1) >= kappalog_2 n refined only by a log_3 n/log_4 n factor from linear forms in logarithms. This follows from Theorem 1.2 (p. 1), a lower bound rad(n^2+1) >= exp(kappa(log_2 n)^2/log_3 n) on the radical. The proof joins estimates from linear forms in logarithms to the author's earlier bounds for the ABC conjecture obtained from Shimura curves: primes with large exponent in n^2+1 are separated from those with small exponent, an elliptic curve is attached to each n, and the Shimura-curve bounds control the number of large-exponent prime divisors. The same technique yields Theorem 1.4 (p. 2), improving the subexponential ABC bounds of the author and of Stewart-Yu to log c <= eta^{-1} exp(kappasqrt((log R) log_2 R)) when a <= c^{1-eta}, and log c <= qexp(kappasqrt((log R) log_2 R)) with R = rad(abc); the paper notes that the second is the first improvement on Stewart and Yu's Theorem 2 in more than two decades. Corollary 1.5 (p. 3) gives P(xy(x+y)) >= kappa(log_2 y)^2/log_3 y, with an absolute kappa>0, as x < y vary over coprime positive integers. Erdos problem 368 asks how large the largest prime factor of n(n+1) is; taking x = 1 and y = n in Corollary 1.5 gives P(n(n+1)) >= kappa*(log_2 n)^2/log_3 n for all large n. The bound for n^2+1 in Theorem 1.1 is a separate result and does not bear on n(n+1).

Source: https://arxiv.org/abs/2312.03566.

Bears on. #368 (lower bound only). Corollary 1.5 at $x=1$, $y=n$ gives $P(n(n+1))\ge\kappa(\log_2n)^2/\log_3n$ for all large $n$, with an absolute $\kappa>0$. The paper does not state this case, and the bound does not determine how large the largest prime factor of $n(n+1)$ is. Theorems 1.1 and 1.2 concern $n^2+1$ and give nothing for $n(n+1)$.

Results to transcribe.

• Theorem 1.1 (p. 1): there is $\kappa>0$ with $P(n^2+1)\ge\kappa(\log_2n)^2/\log_3n$ as $n$ grows, where $P$ is the largest prime factor.
• Theorem 1.2 (p. 1): there is $\kappa>0$ with $\operatorname{rad}(n^2+1)\ge\exp(\kappa(\log_2n)^2/\log_3n)$ as $n$ grows.
• Theorem 1.4 (p. 2): for coprime $a+b=c$ and $R=\operatorname{rad}(abc)$, $\log c\le\eta^{-1}\exp(\kappa\sqrt{(\log R)\log_2R})$ when $a\le c^{1-\eta}$ for a number $\eta>0$, and $\log c\le q\exp(\kappa\sqrt{(\log R)\log_2R})$ with $q=\min\{P(a),P(b),P(c)\}$, each with an absolute $\kappa>0$.
• Corollary 1.5 (p. 3): there is an absolute $\kappa>0$ with $P(xy(x+y))\ge\kappa(\log_2y)^2/\log_3y$ as $x<y$ vary over coprime positive integers; with $x=1$, $y=n$ this bounds $P(n(n+1))$.

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