T-Shirt #5

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Problem Description

A fifth entry in our "Numeric T-Shirt" line:

Back of T-Shirt: "The smallest 3-digit emirp!"

Front: __ __ __ ?

Background & Techniques

A "emirp" is a prime number which forms a different prime number when its digits are reversed. (Palindromic primes are excluded by this definition.)

Not much new work here - we borrowed the IsPrime and Reverse functions from T-Shirt #4 . So all that is left for the SearchBtnClick method is to run through a loop looking for primes, checking if the reverse is a prime, and checking that they are not the same. If a number that meets these three conditions, add it to a display list.

I added a TUpDown box to select how many digits, N, in the emirps of interest. A 3-line IntPower function lets us set the lower limit of our search as 10^(N-1) and the upper limit at one less than 10 times the lower limit. For 3-digit emirps for example, that means that at most we'll search from 10^(3-1)= 10²= 100 up to 10×100-1=1000-1=999. That pretty much covers the 3-digit candidates. We search for emirps up to 9 digits, so we'll limit the display to 100 hits - we really don't need thousands (or millions) of them displayed.

About 50 lines of code here, well within the beginner's range.

Running/Exploring the Program

Suggestions for Further Explorations

This program let's us search for emirps from 2 to 9 digits. The largest value for 32 bit integer types is a little over 2 billion so numbers up to 999,999,999 can be tested. Of course int64 type, 64 bit integers, could test up to somewhere around 10¹⁹ if you would prefer the smallest 19-digit emirp on your shirt. By the way, n binary digits (bits) can represent decimal numbers up to about 0.3n digits. Why? Here's a hint, thanks to the miracle of logarithms:

	10^{? =}2ⁿwhich by taking logarithm of both sides means that
	? × log(10) = n × log(2) and with a little algebra plus our Windows desktop calculator leads to the conclusion that
	? = 0.30103 × n

This is just one of those "reasonableness" checks that can be handy when making quick estimates.

	Why aren't there there any one digit emirps?
	What if we wanted the largest n-digit emirp?

Originally posted: July 27, 2002

Modified:May 18, 2009