Twisty Temperature

Problem:

When Waldo recently did a conversion of a positive integral Celsius temperature $c = 275$ to its Fahrenheit equivalent $f$ (which turned out to be $527$ ), he noticed to his amazement that he could have simply moved the last digit of $c$ to the front to obtain $f$. Doing some intense calculations he failed to discover the next largest such example. Does one exist, and if so, what is it?

Solution:

Let $c = x_{n}\cdot 10^{n-1} + ... + x_{1}\cdot 10^1 + x_{0}$ with $x_{n} > 0$, then $$f = x_{0}\cdot 10^{n-1} + (c-x_{0})/10.$$ We also have that $$f = (9/5)\cdot c + 32.$$ Notice that in order for $f$ to be integral $c$ must be divisible by 5; this implies that $x_0=5$ since it cannot equal 0 (since as a number $f>c$). Our equation then becomes $$(9/5)\cdot c + 32 = 5\cdot 10^{n-1} + (c-5)/10$$ implies $$c = 5\cdot (10^n - 65)/17.$$ Now it turns out that 10 is a primitive root modulo 17, and so it follows that $c$ is integral if and only if $n$ is of the form $16\cdot m + 3$. When $m=0$ we get $c=275$; when $m=1$ we get the next highest such temperature, which is $$5\cdot (10^{19}-65)/17 = 2941176470588235275.$$