# Prime permutations (Euler Problem 49)¶

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https://projecteuler.net/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?

In [20]:

    def sieve_of_eratosthenes(number):
        primes = []
        prospects = [n for n in range(2, number)]
        while prospects:
            p = prospects[0]
            prospects = [x for x in prospects if x % p != 0]
            primes.append(p)
            if p * p > number:
                break
        return primes + prospects

    ps = [p for p in sieve_of_eratosthenes(10000) if p > 1000]

In [21]:

    d = {}
    for p in ps:
        s = "".join(sorted(str(p)))
        try:
            d[s].append(p)
        except KeyError:
            d[s] = [p]

    pss = [value for key, value in d.items() if len(value) >= 3]

In [48]:

    def find_increasing_sequence(xs):
        deltas = [(xs[j] - xs[i], i, j)
            for i in range(0, len(xs) - 1)
            for j in range(i + 1, len(xs))
        ]
        d = {}
        for delta, i, j in deltas:
            if delta in d and d[delta][-1] == i:
                d[delta].append(j)
            else:
                d[delta] = [i, j]
        for delta, sequence in d.items():
            if len(sequence) == 3:
                return [xs[index] for index in sequence]
        return []

    s = [find_increasing_sequence(ps) for ps in pss if find_increasing_sequence(ps)]
    print(s)

    [[1487, 4817, 8147], [2969, 6299, 9629]]

In [49]:

    s = int("".join(map(str, s[1])))
    assert(s == 296962999629)
    print(s)

    296962999629

Congratulations, the answer you gave to problem 49 is correct.

You are the 49654th person to have solved this problem.

Nice work, failx, you've just advanced to Level 2 . 46903 members (5.43%) have made it this far.

This problem had a difficulty rating of 5%. The highest difficulty rating you have solved remains at 5%.

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