For any integer a and c such that $a \gt c$, is it true that at least one integer in the range $[a, a+c]$ will have a prime factor greater than $c$? If this happens to be true, what about if an integer in the range will have a prime factor greater than $2c$? This is for a personal project that I'm working on, but I don't think this could be used in many situations. I'm hoping that there is an easy proof or counterexample for this, or that there is already an existing theorem that directly or indirectly proves this.
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2google Bertrands postulate. Then consider that if $a > c$ so then $a+c > 2c$ – fleablood Feb 18 '21 at 00:21
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@flea, that will show there's a prime between $c$ and $a+c$, but why should it be between $a$ and $a+c$? – Gerry Myerson Feb 18 '21 at 00:42
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@GerryMyerson Where did I ever say it would? – fleablood Feb 18 '21 at 00:50
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1@flea, of course you didn't, but you've left OP to figure out how a prime between $c$ and $a+c$ implies a prime factor exceeding $c$ between $a$ and $a+c$. – Gerry Myerson Feb 18 '21 at 00:56