Find the largest integer $x$ such that $x - 6 | x^{1000} + 2$.

Asked Apr 20 '24 at 23:37

Active Apr 20 '24 at 23:46

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Find the largest integer $x$ such that $x - 6 | x^{1000} + 2$.

How would I go about solving for $x$? I am told that the answer should have just under 800 digits, so brute forcing is not possible. I know there are theorems that I can use, but I am not certain how to apply them to this problem.

edited Apr 20 '24 at 23:46

Bill Dubuque

282,220

asked Apr 20 '24 at 23:37

Karl Stolze

1

Write $x^{1000}=((x-6)+6)^{1000}$ – lhf Apr 20 '24 at 23:42
See e.g. here in the linked dupe for general methods. – Bill Dubuque Apr 20 '24 at 23:48

Find the largest integer $x$ such that $x - 6 | x^{1000} + 2$.

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