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 

bulletBrowse source code
bulletDownload source 
bulletDownload  executable

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? = 2n  which 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
 

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