Edwin asks: << . . . . . . Does every positive integer K appear in some square? That is, given K are there integers x,y and N such that N^2 = xKy? Integers may be expressed in any base and xKy is concatenation of x, K and y.
Although I posted some vague mumblings addressing this yesterday, Rich later pointed out in conversation that that this amounts to nothing more than the existence of a square root. Here's one way to look at it: Given K as below and even given x as well, place the concatenation xK immediately after a decimal point to get a fraction .xK . Now just expand sqrt(.xK) to enough decimal places -- call it s = trunc(sqrt(.xK)) -- such that s^2 after the decimal point begins with xK . Then multiply everything by a high enough power of 10 so that this works with integers. Let S = (10^p)s so that S is an integer; then S^2 will have initial string xK . --Dan