Fermat's and Trial Division
Consider trying to factor the prime number N=2345678917, but also compute b and a-b throughout. Going up from, we can tabulate:
| a: | 48433 | 48434 | 48435 | 48436 |
| b2: | 76572 | 173439 | 270308 | 367179 |
| b: | 276.7 | 416.5 | 519.9 | 605.9 |
| a-b: | 48156.3 | 48017.5 | 47915.1 | 47830.1 |
In practice, one wouldn't bother with that last row, until b is an integer. But observe that if N had a subroot factor above, Fermat's method would have found it already.
Trial division would normally try up to 48432; but after only four Fermat steps, we need only divide up to 47830, to find a factor or prove primality.
This all suggests a combined factoring method. Choose some bound ; use Fermat for factors between and . This gives a bound for trial division which is . In the above example, with the bound for trial division is 47830. A reasonable choice could be giving a bound of 28937.
In this regard, Fermat's method gives diminishing returns. One would surely stop before this point:
| a: | 60001 | 60002 |
| b2: | 1254441084 | 1254561087 |
| b: | 35418.1 | 35419.8 |
| a-b: | 24582.9 | 24582.2 |
Read more about this topic: Fermat's Factorization Method
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