Euler problem 49

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

What 12-digit number do you form by concatenating the three terms in this sequence?

analyse:
start with next odd number 1489 and increment by two.

Problem49 =296962999629 elapsed time:    2 ms. Test Passed.

Euler problem 41

We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is also prime.

What is the largest n-digit pandigital prime that exists?

analyse:


The 8 and 9 digit pandigital 987654321 are divideble by 3 (Sum digits divide evenly by 3).
and thus are composit numbers and no primes.
So we start with a 7 digit padigital number 7654321.


Performance improvements:


1) Use permutation to generate pandigitals. This reduces loop from 6419754 to 5040. 
2) Use the fast Miller-Rabin test to check if a big number is a prime.

Problem41 =     7652413 elapsed time:    2 ms. Test Passed.

Euler problem 24

A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:

012   021   102   120   201   210

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

Problem24 =  2783915460 elapsed time:    0 ms. Test Passed.