More on squaring and multiplying large integers

Methods of squaring and multiplying large integers are discussed. The obvious O(n²) methods turn out to be best for small numbers. Existing O(n^{log 3}log/ ²)≈O(n^1.585) methods become better as the numbers get bigger. New methods that are O(^log5log/ ³ )≈0(n^1.465), O(n^{log 7}log/ ⁴)≈O(n^1.404), and O(n^{log 9}log/ ⁵)≈O(n^1.365) presented. In actual experiments, all of these methods turn out to be faster than FFT multipliers for numbers that can be quite large (>37,000,000 bits). Squaring seems to be fundamentally faster than multiplying but it is shown that T_multiply⩽2T_square+O(n)