Palindromic Patterns of Greatest Divisors $\leq k$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T11:16:16Z http://mathoverflow.net/feeds/question/119447 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119447/palindromic-patterns-of-greatest-divisors-leq-k Palindromic Patterns of Greatest Divisors $\leq k$ Fred Kline 2013-01-21T06:59:31Z 2013-01-21T11:11:33Z <p>We work with the positive integers $\in \mathbf{Z^{+}}$.<br> The list length is the least common multiple for all numbers $\leq k$: $$\text{rlcm}(1)\equiv 1; \text{rlcm}(2)\equiv 2; \text{rlcm}(n) \equiv \text{lcm}(\text{rlcm}(n - 1), n)$$ The greatest divisor $\leq k$ is a definition that gives a unique identity to the $1$ (one) because $1$ is the greatest divisor $\leq k$ for numbers that are co-prime to all values $\leq k$. The greatest divisor routine: <code>$$\text{maxd}(1,n)\equiv1;\text{maxd}(k,n)\equiv \begin{cases} k &amp;\text{for }0= n \text{mod} k \\ \text{maxd}(k-1,n) &amp;\text{for } 0\neq n \text{mod} k \end{cases}$$</code></p> <p>List of greatest divisors $\leq k=4$. List length is lcm$(1,\dots,k)$.<br> $1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4$</p> <p>Five replications of the above list:<br> $1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3,$<br> $1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4$</p> <p>List of greatest divisors $\leq k=5$. List length is lcm$(1,\dots,k)$. $1, 2, 3, 4, 5, 3, 1, 4, 3, 5, 1, 4, 1, 2, 5, 4, 1, 3, 1, 5, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5,$<br> $1, 4, 3, 2, 5, 4, 1, 2, 3, 5, 1, 3, 1, 4, 5, 2, 1, 4, 1, 5, 3, 4, 1, 3, 5, 4, 3, 2, 1, 5$</p> <p>List of greatest divisors $\leq k=6$. List length is lcm$(1,\dots,k)$. $1, 2, 3, 4, 5, 6, 1, 4, 3, 5, 1, 6, 1, 2, 5, 4, 1, 6, 1, 5, 3, 2, 1, 6, 5, 2, 3, 4, 1, 6,$<br> $1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 4, 3, 2, 1, 6$</p> <p>When you remove the last element of each list it will read the same in both directions.<br> Note-1: the first $k$ numbers of each list are the divisors $\leq k$.<br> Note-2: there is at least one $1$ between each $k$ in its respective list. (i.e., a co-prime in every segment of size $k$). </p> <p>How can I prove that these finite lists repeat forever?<br> How can I prove that there is at least one $1$ in each segment of size $k$?</p> http://mathoverflow.net/questions/119447/palindromic-patterns-of-greatest-divisors-leq-k/119457#119457 Answer by Gerry Myerson for Palindromic Patterns of Greatest Divisors $\leq k$ Gerry Myerson 2013-01-21T11:11:33Z 2013-01-21T11:11:33Z <p>Let $k=13$. The segment of length $17$ from $2184$ to $2200$, inclusive, has no $1$; every integer in that range has a prime divisor not exceeding $13$. </p>