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Let $p \ge 5$ be a prime number. Find the largest length of an arithmetic progression, of positive ratio, of positive integers whose terms do not contain the digit $1$ in their p-adic expansion.

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I suspect that ratio should be read as difference. If so, you’re being asked to determine the greatest possible length of a strictly increasing arithmetic sequence of positive integers, whose base $p$ representations do not contain the digit $1$. In base five, for instance, if you start out with $2,4$, you’re blocked, because $6=11_{\text{five}}$. If you start at $12=22_{\text{five}}$ with a common difference of $3$ you can do a little better: $22_{\text{five}},30_{\text{five}},33_{\text{five}}$. But the next number in the sequence is $41_{\text{five}}$, so you’re blocked again. And be starting at $14=24_{\text{five}}$, I can do better yet: $24_{\text{five}},32_{\text{five}},40_{\text{five}},43_{\text{five}}$. (I’ll leave it to you to verify that I’m blocked again.)

HINT: Look at the $1$’s digit, the least significant digit. Say that your sequence is $a,a+d,a+2d$, $a+3d$, and so on. If $b\ne 0$ is the $1$’s digit of $d$, what can you say about the numbers $d,2d,3d,\dots,pd$ modulo $p$? And you don’t have to worry about a difference $d$ that ends in $0$: the $1$’s digit remains constant throughout your sequence, and you might as well just throw it away. E.g., instead of looking at $242_{\text{five}},322_{\text{five}},402_{\text{five}},432_{\text{five}}$, with difference $d=30_{\text{five}}$, look at $24_{\text{five}},32_{\text{five}},40_{\text{five}},43_{\text{five}}$, with difference $d=3$.

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    let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/6037/discussion-between-finial-and-brian-m-scott)2012-10-05