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• Paul Rutherford

# PERFECT DAY

It's a perfect day.

That's not an emotional or meteorological statement: I had to do the weekly shop this morning. And the rising temperature here has cut short the gardening.

But mathematically, today is as good as it gets.

A perfect number is an integer greater than zero which - when all its factors are added together - gives that integer.

The first perfect number is 6. It's divisible by 1, 2 and 3. 1+2+3=6.

The next perfect number is 28, because 1+2+4+7+14=28.

Hence today, 28/6 (or 6/28 for US readers), is the calendar's perfect day.

As you've probably guessed, perfect numbers aren't thick on the ground. If you were hoping that we'd get another perfect day in the year, you'r going to be disappointed. The next perfect number is 469 (1+2+4+8+16+31+62+124+248).

The one that is 8128 (so make a note in your diary for 8 Jan or 1 Aug 2028, depending which side of the Atlantic you live).

As evidence of their rarity, the fifth known perfect number is 33,550,336.

So far, mathematicians have identified 37 perfect numbers, most of which have been calculated in the past 20 years, thanks to super-computing.

Interestingly, despite the fact that they are all divisible by 1, so far no odd perfect number has been found.

Not surprising, as it has been proved that any odd perfect number must exceed 10300 and must be divisible by a prime power exceeding 1020.