Timeline for Squareful values of polynomials
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 8, 2017 at 15:52 | comment | added | joro | @DanielLoughran Here is selfcontained proof. Fix $n>2$, let $F(x)=x^n+1$. Assume $F(X)$ is squareful $F(X)=u^2v^3$. Take the abc triple $a=X^n,b=1,C=X^n+1=u^2v^3$. $rad(abc)=xuv=O(X^{n/2+1})$ while $c \sim X^n$. | |
| May 8, 2017 at 15:43 | vote | accept | Daniel Loughran | ||
| May 8, 2017 at 15:25 | comment | added | joro | @DanielLoughran $rad$ is the radical, the product of the distinct prime factors. The radical in the RHS is at most $|n|^{\deg(F)/2}$. The LHS is $|n|^{\deg(F)-1-\epsilon}$. If $\deg(F)>2$ the inequality doesn't hold, contradicting abc. | |
| May 8, 2017 at 14:38 | comment | added | Daniel Loughran | I don't quite follow. Could you provide more details how the conjecture of Langevin helps with my question? Are you also able to give precise references for the claims at the beginning and the end of your answer? | |
| May 8, 2017 at 10:54 | history | answered | joro | CC BY-SA 3.0 |