There are about 18.446 quintillion more 64-bit integers than 32-bit integers.
replyTrue, but there are as many 64-bit integers as pairs of 32-bit integers.
replyTherefore the fact that relatively few 64-bit numbers are products of 32-bit integers means that a lot of pairs of 32-bit integers give by multiplication the same product.
I think they meant to write "There are about 4 billion TIMES more 64 bit integers than 32 bit integers".
replyIndeed, edited the mistake
replyThere are about 2^64 more 64-bit integers than 32-bit integers.
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replyThe chance of a random 64-bit integer matching some pair of 32-bit integers is a 100%, though.
replyOr, the odds of a random 64-bit integer being a 32-bit integer are the same as you or me guessing a random 32 bit integer.
replyWonder what the limit is as you add more 32 bit integers to the product. Just the primes over 32 bit?
replyIf you're allowed to multiply as many 32-bit numbers as you want, the only numbers you won't be able to achieve by so doing are those with any prime factor larger than 2^32.
replyThis is more than just the prime numbers. For example, a 41-bit prime can be multiplied by 16 and it will still fit into 64 bits.
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replyWhat are you assuming about overflow? Three 32-bit numbers multiply out to 96 bits.
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