Project Euler problem 49

Prime permutations

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.

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Project Euler problem 47

Distinct primes factors

The first two consecutive numbers to have two distinct prime factors are:

        14 = 2 × 7
        15 = 3 × 5

The first three consecutive numbers to have three distinct prime factors are:

        644 = 2² × 7 × 23
        645 = 3 × 5 × 43
        646 = 2 × 17 × 19.

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Project Euler problem 46

Goldbach’s other conjecture

It was proposed by Christian Goldbach that every odd composite number can be written as the sum of a prime and twice a square.

        9  = 7  + 2×1^2
        15 = 7  + 2×2^2
        21 = 3  + 2×3^2
        25 = 7  + 2×3^2
        27 = 19 + 2×2^2
        33 = 31 + 2×1^2

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