"Unfortunately, we can’t do that here – a 64-bit integer only has so many possible values" and "The remaining 12 or 13 bits give us the extra “digits” we need f

"Unfortunately, we can’t do that here – a 64-bit integer only has so many possible values" and "The remaining 12 or 13 bits give us the extra “digits” we need for preventing carries." using 64 bit registers, but only using 51 bits, has no overflow. Using 64 bit adds in 64 bit registers overflows, making the carry updates more costly.