Formula for Calculating Squares Starting with n

Hey guys! I was pretty bored today and I happened to have my calculator on me. And for some reason, this problem was on my mind.

So, I started to think about how it gets solved and such and wanted to generalize some formula that could find all the squares starting with xx. And I found something! Here is what I ended up with:

10nx\left\lceil { 10 }^{ n }\sqrt { x } \right\rceil

So, for a given xx, it would output a number you'd have to square to get a perfect square starting with xx.

For example:

  1. Perfect squares starting with 8888:

1028888=9428(94282=88887184)1038888=94277(942772=8888152729)1048888=942762(9427622=88880018864) \left\lceil { 10 }^{ 2 }\sqrt { 8888 } \right\rceil =9428\quad ({ 9428 }^{ 2 }=88887184)\\ \left\lceil { 10 }^{ 3 }\sqrt { 8888 } \right\rceil =94277\quad ({ 94277 }^{ 2 }=8888152729)\\ \left\lceil { 10 }^{ 4 }\sqrt { 8888 } \right\rceil =942762\quad ({ 942762 }^{ 2 }=88880018864)\\

(so on, n would increase by 1 each time...)

  1. Perfect squares starting with 987654321:

105987654321=3142696806(31426968062=9876543214442601636)106987654321=31426968053(314269680532=987654321004282610809) \left\lceil { 10 }^{ 5 }\sqrt { 987654321 } \right\rceil =3142696806\quad ({ 3142696806 }^{ 2 }=9876543214442601636)\\ \left\lceil { 10 }^{ 6 }\sqrt { 987654321 } \right\rceil =31426968053\quad ({ 31426968053 }^{ 2 }=987654321004282610809)

(so on...)

My question is this: See the xx in that formula that I stated at the start of this and never went on to define? That's the thing, I don't know how to define it, as in without guessing and checking, I don't know the smallest n for which the result is valid. I know from playing around that n depends in some way on the amount of digits of x and the parity of the x. If x is even, the smallest n for which the formula is correct will be even (vice versa for odd). Also, the larger x is, the larger n seems to have to be in order for it to hold.

I've tried many things but I can't seem to find out how to determine the smallest value n needed for the formula to carry out properly.

Could anyone provide some insight? It would be much appreciated. :)

This is just for fun!

Note by Andrew Tawfeek
3 years, 4 months ago

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Have you read the solution to the problem? it should be pretty clear that I'm not just randomly testing for the value of nn.

Calvin Lin Staff - 3 years, 4 months ago

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I understood slightly the differences between when (referring to the solution there) N is odd/even how it would affect the value of nn, but I can't seem to understand what decides the lowest value of nn.

Andrew Tawfeek - 3 years, 4 months ago

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Moving this into the solution discussion of the problem directly.

Calvin Lin Staff - 3 years, 4 months ago

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