square number?
Can a number 44…41, whose decimal representation consists of an odd number of digits 4 followed by a digit 1, be a square number?
This is problem 2 of the first round of this year's National Mathematics Competition. Today (March 4) is the deadline for submissions, so the post office seems to have closed, so it's probably okay to talk about it online (if you haven't solved this problem yet, you probably haven't solved any of the others either).
In my proof I show that the 2n-digit number described above lies between the squares of 66..66 and 66..67 (each with n digits) and therefore cannot itself be a square number.
Are there other working proof ideas?
And: Why can’t you rehire?
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I mean, this question would have been swung through the forum a few weeks ago. It was done with a simple modulo layout – probably contrary to the intention of the author – with the number 4en being completely irrelevant. Could have been modulo 100, maybe you’ll find the question.
I haven’t found on GF.
If the number of 4en before the 1 is odd, it cannot be a square number. For an even number of 4, at least 441 = 212 is an example of a square number.
I was wrong, went to 66….61, sorry,
That’s much easier:
Life can be so easy. Thank you.
Although my evidence indicated in the question also works, it is much more complicated to write up and much less elegant.
Maybe if you go mod 11, with odd number 4en the rest always seems to be equal to 8 that is not a square rest mod 11.
This is surprisingly similar to the competition task.
Yes, it works at 6..661, but unfortunately not at 4…441