Today I turned 24505 days old and was surprised to find the following information about this number:
In all the time that I've been analysing my daily numbers, this was the first time that I'd seen the number being representable by the sum of the squares of six different pairs of numbers. Similarly, I'd not seen a number being the hypotenuse of four different Pythagorean triples. It lead me to think whether this was a record of some sort.
Well, a little research shows that this is certainly not the case. For example, the first numbers that are representable as the sum of the squares of six different pairs of numbers are:
5525, 9425, 11050, 12025, 12325, 13325, 14365, 15725, 17225, 17425, 18785, 18850, 19825, 21125, 22100, 22525, 23725, 24050, 24505, 24650, 25925, 26650, 26825, 27625, 28730, 28925, 29725, 31025, 31265, 31450, 31525, 32045, 32825, 34450, 34645, 34850
There are numbers that are representable as the sum of the squares of seven different pairs of numbers but the first such number is 105625. Numbers that are representable as the sum of the squares of eight different pairs of numbers are interestingly far more common than seven and begin thus:
27625, 32045, 40885, 45305, 47125, 55250, 58565, 60125, 61625, 64090, 66625, 67405, 69745, 77285, 78625, 80665, 81770, 86125, 87125, 90610, 91205, 94250, 98345, 98605, 99125, 99905, 101065, 107185, 110500, 111605, 112625, 114985, 117130, 118625
71825, 93925 and 122525 are the first three numbers that are representable as the sum of the squares of nine different pairs of numbers. Of numbers that are representable as the sum of the squares of ten different pairs of numbers, the first is 138125. The OEIS doesn't give any results for 11 onwards but there's no reason to not suppose that numbers exist that are representable as the sum of the squares of eleven different pairs of numbers and beyond.
Let's summarise the results:
In all the time that I've been analysing my daily numbers, this was the first time that I'd seen the number being representable by the sum of the squares of six different pairs of numbers. Similarly, I'd not seen a number being the hypotenuse of four different Pythagorean triples. It lead me to think whether this was a record of some sort.
Well, a little research shows that this is certainly not the case. For example, the first numbers that are representable as the sum of the squares of six different pairs of numbers are:
5525, 9425, 11050, 12025, 12325, 13325, 14365, 15725, 17225, 17425, 18785, 18850, 19825, 21125, 22100, 22525, 23725, 24050, 24505, 24650, 25925, 26650, 26825, 27625, 28730, 28925, 29725, 31025, 31265, 31450, 31525, 32045, 32825, 34450, 34645, 34850
There are numbers that are representable as the sum of the squares of seven different pairs of numbers but the first such number is 105625. Numbers that are representable as the sum of the squares of eight different pairs of numbers are interestingly far more common than seven and begin thus:
27625, 32045, 40885, 45305, 47125, 55250, 58565, 60125, 61625, 64090, 66625, 67405, 69745, 77285, 78625, 80665, 81770, 86125, 87125, 90610, 91205, 94250, 98345, 98605, 99125, 99905, 101065, 107185, 110500, 111605, 112625, 114985, 117130, 118625
71825, 93925 and 122525 are the first three numbers that are representable as the sum of the squares of nine different pairs of numbers. Of numbers that are representable as the sum of the squares of ten different pairs of numbers, the first is 138125. The OEIS doesn't give any results for 11 onwards but there's no reason to not suppose that numbers exist that are representable as the sum of the squares of eleven different pairs of numbers and beyond.
Let's summarise the results:
- 5
- 65
- 325
- 1105
- 8125
- 5525
- 105625
- 27625
- 71825
- 138125
I won't deal here with the numbers n for which n^2 can be represented as the sum of squares of different pairs of numbers, in other words n forms the hypotenuse of a Pythagorean triple. However, a similar analysis is possible.
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