​How to find square numbers?

[Skip Cave]

Question: is 21 ^ 21 a square number?

Test: Take the square root of 21^21 and test to see if the result is a integer (=>.) (is the floor of the result the same as the result)

21x^21

5842587018385982521381124421

Take the square root of 21^21, store in n, and list it:

]n=.%:21x^21

7.64368e13

Is n an integer?

(=>.) n

1

Integer test says so.

Square n to get back to 21x^21:

x: n^2

5842587018385982340114415616

However,

21x^21 is

5842587018385982521381124421

Compare the two numbers:

5842587018385982340114415616

5842587018385982521381124421

Not the same.

21x^21 is not a square, even though the integer test (=>.) says so.

What went wrong? How can one test for large square numbers?

Skip

Skip Cave
Cave Consulting LLC

[Elijah Stone]

first problem:

(=>.)n
1
(=!.0>.)n
0

second problem: all floating-point numbers above 2^52 are whole numbers, so
for any number above (2^52)^2, the test (even amended for comparison
tolerance) will always claim it to be square. (for numbers below that, i
_believe_ the test is correct, assuming %: is correctly rounded on exact
numbers, which i'm not sure of.)

the correct way to do it is to avoid introducing inexact results in the first
place; <.@%: will do this (<.%:y will not!). hence i would do something like:

(= *:@(<.@%:)) 21x^21
0

also, x:n^2 is not right in general (not necessarily wrong, but if you find
yourself writing that it is _probably_ wrong); (x:n)^2 is much likelier to do
what you want most of the time

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[Henry Rich]

x: and *: do not commute. 

Henry Rich