The initial impetus for this project! Factoring the digits of π (3.14159...) as decimal expansion 3, 31, 314, 3141, 31415, 314159, ... also known as OEIS A078604.
The first 200 digits of π were completed in 2023, with the final stragglers submitted by ejeancolas.
As of early 2024 there are 53 items in the sequence below 314 digits that remain unfactored. ECM curves 1-8 have been thoroughly covered. Curve 9 has been run to spec (see Curves below) but hasn't yet been exhausted. Curves 10-13 are in progress and have yielded some remaining factors.
The RSA Numbers were added in 2020, shortly before RSA-232 and RSA-250 were announced.
A friend told me about Home Primes and that he was working on HP49(119) around the same time, so I added that to the queue alongside RSA and π. It has apparently eluded factorization for almost a decade since its discovery at this point, and has burned through most / all of the Curves below several times over.
Based on curves from Mersenne Wiki "Elliptic Curve Method"
| Digits | B1 | GMP-ECM B2 | Curves |
|---|---|---|---|
| 15 | 2,000 | 147,396 | |
| 20 | 11,000 | 1,873,422 | 86 |
| 25 | 50,000 | 12,746,592 | 214 |
| 30 | 250,000 | 128,992,510 | 430 |
| 35 | 1,000,000 | 1,045,563,762 | 910 |
| 40 | 3,000,000 | 5,706,890,290 | 2,351 |
| 45 | 11,000,000 | 35,133,391,030 | 4,482 |
| 50 | 43,000,000 | 240,490,660,426 | 7,557 |
| 55 | 110,000,000 | 776,278,396,540 | 17,884 |
| 60 | 260,000,000 | 3,178,559,884,516 | 42,057 |
| 65 | 850,000,000 | 15,892,628,251,516 | 69,471 |
| 70 | 2,900,000,000 | 105,101,237,217,912 | 102,212 |
| 75 | 7,600,000,000 | 425,332,376,469,022 | 188,056 |
| 80 | 25,000,000,000 | 2,551,982,328,195,322 | 265,557 |
- Start with a composite
X = N * MwhereN ≅ MandD = log10(N) - Look up
Din theDigitscolumn to find your row - If your
Dis between rows, either pick the larger value or pick a number between the two values - Find the
B1value andCurvescount
With these, you have the parameters for ecm:
echo ${X} | ecm -one -c ${CURVES} ${B1}
ECM will set the B2 parameter appropriately to match these table values, or you can specify it as
echo ${X} | ecm -one -c ${CURVES} ${B1} ${B2}
If N ≫ M you should use D = log10(M) to find the smaller factor more efficiently.