Perfect square problem!

[Guest wmc]

Show that a number of the form

a00..00b

 

with at least one zero,

and a in the range 1,2,...,9

and b is 1,4, or 9

cannot be a perfect square

 

Help please!!

[Algebracus]

An easy, but perhaps a little unsophisticated approach is to check perfect squares modulo 100. Since (50n + k)^2 and (50n - k)^2 are both congruent to k^2 for every natural numbers n and k, it is enough to check the numbers 0, 1, 2, 3, ..., 25. We can of course check all these 25 numbers, but since we are only interested in the endings 01, 04, 09, we check only the numbers that gives perfect squares ending in 1, 4 or 9, that is, 1, 2, 3, 7, 8, 9, 11, 12, 13, 17, 18, 19, 21, 22 and 23. The endings of these are 1, 4, 9, 49, 81, 21, 44, 69, 89, 44, 24, 61, 41, 84 and 29, none of them being among 01, 04 or 09, giving the desired result.

 

The result is stronger than what you asked for, but that is not exactly a loss.

[Martinez]

My question would ave to be:

 

If no such thing as the perfect circle, how possibly the perfect square?

[uncool]

Martinez:

A perfect square is a way in number theory to denote any number that is equal to some integer times itself. For example, 4 is a perfect square because it is equal to 2*2.

The impossibility of a perfect circle and a perfect square in reality is not because of mathematics, but because of physics and things like atoms and quantum physics.

 

Algebracus, notice that the ending of 51^2 = 2601, 52^2 = 2704, etc.

 

My proof:

Let there b c 0's.

The square root must end in 0000...1, 2, or 3 (otherwise, the ending of the square could not be 000...0b). The square would therefore have 3 parts: the square of the part before the 0, the geometric mean of the two parts, and the part after the 0's. Therefore, you can't have only 0's in the middle.

For example:

1000001^2 = 1000002000001 - the 3 parts are the 1 at the beginning, the 2 in the middle, and the 1 at the end.

-Uncool-

[Martinez]

Thanks much, uncool...and from one of my kids as well. Your answer happened to coincide with a homework assingment. My query still is unresolved from a theoretical standpoint - If no perfect circle, then no perfectly straight line either - so how to describe the perfect square?? Fun for me to think on. The answer I suspect is in knowing how to account for oblateness of the circle.