Take any odd number greater than 3.
Square the number.
Subtract 1.
The resulting number is divisible by 24.
Always.
I do no have any proof for this. But I have tried it for many odd numbers big and small. All have the same property.
My father says he has not heard of this property before. So perhaps I should call it Abhinav's Conjecture. :)
Let's try a few example:
19 x 19 - 1 = 360 --- divisible by 24
31 x 31 - 1 = 960 --- divisible by 24
331 x 331 - 1 = 109560 --- divisible by 24
3571 x 3571 - 1 = 12752040 --- divisible by 24
10093 x 10093 - 1 = 101868648 --- divisible by 24
Hopefully, this was interesting to my viewers.
Square the number.
Subtract 1.
The resulting number is divisible by 24.
Always.
I do no have any proof for this. But I have tried it for many odd numbers big and small. All have the same property.
My father says he has not heard of this property before. So perhaps I should call it Abhinav's Conjecture. :)
Let's try a few example:
19 x 19 - 1 = 360 --- divisible by 24
31 x 31 - 1 = 960 --- divisible by 24
331 x 331 - 1 = 109560 --- divisible by 24
3571 x 3571 - 1 = 12752040 --- divisible by 24
10093 x 10093 - 1 = 101868648 --- divisible by 24
Hopefully, this was interesting to my viewers.
Abhinav, please note that primes > 2 are always odd. So they can be written as 2n+1 where n is any integer. Square this and you get (using the binomial formula): (2n)^2 + 2*(2n)*1 + 1^2. Subtracting 1 and multiplying out the rest gives 4(n^2)+4n. This can be written as 4(n^2+n), a number that is always dividable by 4. The property you found not only applies to all odd primes, but to all odd numbers - take 15*15-1 = 225 - 1 = 224 wihch is also dividable by 4.
ReplyDelete@ein echter Heiner,
ReplyDeleteDuly noted.
I hadn't thought of this approach
before. Clever, though.