## A BIG number

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These numbers have the property that if you double them you get a perfect square

2, 8, 18, 32, 50, 72, 98, ...

And these have the property that if you multiply them by 3 you get a perfect cube

9, 72, 243, 576, 1125, ...

Some numbers are in both lists; the smallest is 72 and

2 x 72 = 122, 3 x 72 = 63.

It's not possible to extend this and find a number n such that 2n is a square, 3n a cube and 4n a fourth power. However it was asked on Quora what the smallest number is such that n such that 2n is a square, 3n a cube and 5n a fifth power. Alon Amit showed the answer is 215320524. In full this is

6810125783203125000000000000000.

I decided to extend this to the next prime number, 7, and found the smallest n such that 2n is a square, 3n a cube, 5n a fifth power and 7n a seventh power is 21053140584790. This impressive number evaluates to

150462810922326152710290228433686961530697356776074449373600141938371053848189980134027578261857302770024765419887333164323078738017254430529707573248000000000000000000000000000000000000000000000000000000000000000000000000000000000000

which has 234 digits.

I think we can continue and add 11, 13 etc. but at this point I ran out of steam!

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