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 \[ 5\cdot (10^{19}-65)/17 = 2941176470588235275.\]

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