TITLE: Minimum number of integer-sided squares needed to tile an $m$ by $n$ rectangle.
QUESTION [16 upvotes]: Let $T(m,n)$ for integers $m,n$ be the least number of integer-sided squares needed to tile an $m\times n$ rectangle. Clearly $T(kx,ky)\leq T(x,y)$. Are there integers $x,y,k\gt 1$, such that $T(kx,ky)<T(x,y)$?

REPLY [2 votes]: Here's the Bouwkamp code of two simple perfect squared rectangles:

13: 304x274 (141,78,85)(71,7)(92)(133,8)(51,28)(23,97)(74);
15: 152x137 (83,69)(31,38)(54,29)(16,15)(8,30)(25,4)(1,22)(21).

There is no lower-order simple squared rectangle of either size, but there might be compound ones. If not then T(304,274) = 13, T(152,137) = 15, and T(304,274) < T(152,137).