The Erdős-Moser equation revisited using continued fractions
See Yves Gallot, Pieter Moree, Wadim Zudilin,
The Erdős-Moser equation 1k + 2k + ... + (m−1)k = mk revisited using continued fractions.
Math. Comp. 80 (2011), 1221–1237.
On September 2024, Robert Gahan established the new benchmark m > 101010 with cfErdosMoser. The processor was an Intel® Core™ i9-13980HX with 64GB RAM, running Windows 11 Pro. It took about 61 hours with 8 threads and the peak memory usage was 33 GB to prove, using N = 3 · 19#, that conditions (a), (b), (c) are not satisfied if qj < 7.76 · 10 10 381 635 903. With N = 28 · 35 · 54 · 7, it took about 91 hours and the peak memory usage was 56 GB to find the solution qj = 7.69 · 10 16 771 044 760 that fits conditions (a), (b), (c) and partly (d).
Theorem 3 (2024). If an integer pair (m, k) with k ≥ 2 satisfies the Erdős-Moser equation, then m > k > 101010.
This new version of the program is not based on the binary expansion of log 2. Instead, the main diagonal Padé approximant to log 2 at the point one is calculated. It can be computed using a generalized continued fraction but then convergents are not irreducible fractions. However, the canonical form can be obtained by dividing the numerator and denominator by a known function.
If the normalized convergents of the generalized continued fraction and the coefficients of the regular continued fraction are evaluated together then the size of the remaining j th and (j + 1)th fractions is smaller than the size of the binary expansion needed for evaluating j coefficients. Because memory size is the main limitation of this computation, this method is more efficient.
If qj is a n-digit number, memory usage is about 4n.
The following table provides the smallest integers j satisfying conditions (a), (b) and (c) of Theorem 2 and (d) is checked for p ≤ 43 such that 3 is a primitive root modulo p.
n# is the primorial function.
The solutions to 28 · 35 · 54 · 7 = 272160000 and 28 · 35 · 54 · 72 = 1905120000, found by Robert Gahan, are the best known results.
23# = 223092870 is the most wanted candidate.
| N | jN | aj+1 | qj (rounded down) | qj mod 6 | p(qj) |
|---|---|---|---|---|---|
| 23# | > 78 · 10 9 | > 10 40 000 000 000 | |||
| 28 · 35 · 54 · 72 | 32 544 936 570 | 457 913 312 295 | 7.699688 · 10 16 771 044 760 | -1 | |
| 28 · 35 · 54 · 7 | 13 988 361 422 | 107 695 241 629 | -1 | 7 | |
| 32 544 869 308 | 65 416 187 470 | 7.699688 · 10 16 771 044 760 | -1 | ||
| 3 · 19# | 20 145 968 414 | 8 176 658 150 | 7.766616 · 10 10 381 635 903 | +1 | |
| 2 · 19# | 13 078 292 184 | 5 890 259 269 | 6.066802 · 10 6 739 544 561 | -1 | |
| 19# | 13 050 765 960 | 4 245 878 986 | 1.202797 · 10 6 725 335 083 | -1 | |
| 11! | 5 151 307 662 | 14 371 269 642 | 1.086746 · 10 2 654 644 376 | -1 | |
| 28 · 35 · 54 | 2 015 385 392 | 21 941 638 090 | -1 | 19 | |
| 3 236 257 942 | 11 535 576 954 | 5.427816 · 10 1 667 658 416 | +1 | ||
| 28 · 35 · 53 | 2 015 279 170 | 4 388 327 617 | -1 | 19 | |
| 3 236 170 820 | 2 307 115 390 | 5.427816 · 10 1 667 658 416 | +1 | ||
| 10! | 1 738 197 230 | 3 345 301 699 | +1 | 5 | |
| 2 579 384 226 | 1 116 890 723 | 5.498581 · 10 1 329 206 901 | +1 | ||
| 28 · 35 · 52 | 1 738 154 976 | 1 433 700 727 | +1 | 5 | |
| 1 977 626 256 | 853 324 651 | 1.196828 · 10 1 019 133 881 | -1 | ||
| 17# | 367 396 948 | 127 090 830 | -1 | 5 | |
| 586 036 322 | 331 038 577 | +1 | 19 | ||
| 639 808 912 | 675 894 301 | 2.276299 · 10 329 726 891 | +1 | ||
| 28 · 35 · 5 | 351 661 538 | 85 898 211 | 9.729739 · 10 181 214 202 | -1 | |
| 28 · 35 | 155 830 946 | 31 664 035 | 2.257099 · 10 80 303 211 | -1 |