**Sylvester's sequence** is an integer sequence (s_{i}) where s_{0} is 2,
and s_{i} is one plus the product of all i previous terms:
s_{i} = s_{0}s_{1}...s_{i-1} + 1.

The sequence begins 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443.

The empty product of no terms is 1, so s_{0} = 1 + 1 = 2 is a natural start.

It can easily be proved by induction that
s_{i+1} = s_{i} × (s_{i}−1) + 1. This reminds of the
Fermat numbers F_{i} = 2^{2i}+1, where:

F_{0} = 3, F_{i} = F_{0}F_{1}...F_{i-1}
+ 2, F_{i+1} = (F_{i}−1)
× (F_{i}−1) + 1.

The numbers in Sylvester's sequence are relatively prime. Most primes do not divide any
of them. It is not known whether all the numbers are square-free. No prime
factor at this site has a square that also divides the number.

The following table shows known factors up to s_{22}.
P*n* is an *n*-digit
prime. C*n* is an *n*-digit composite number with unknown
factorization.

i |
Factorization of s |
---|---|

0 | 2 (prime) |

1 | 3 (prime) |

2 | 7 (prime) |

3 | 43 (prime) |

4 | 13 × 139 |

5 | 3263443 (prime) |

6 | 547 × 607 × 1033 × 31051 |

7 | 29881 × 67003 × 9119521 × 6212157481 |

8 | 5295435634831 × 31401519357481261 × 77366930214021991992277 |

9 | 181 × 1987 × 112374829138729 × 114152531605972711 × P68 |

10 | 2287 × 2271427 × 21430986826194127130578627950810640891005487 × P156 |

11 | 73 × C416 |

12 | 2589377038614498251653 × 2872413602289671035947763837 × C785 |

13 | 52387 × 5020387 × 5783021473 × 401472621488821859737 × 287001545675964617409598279 × C1600 |

14 | 13999 × 74203 × 9638659 × 57218683 × 10861631274478494529 × C3293 |

15 | 17881 × 97822786011310111 × 54062008753544850522999875710411 × C6618 |

16 | 128551 × 115220560101116343072340969000241209 × C13300 |

17 | 635263 × 1286773 × 21269959 × C26661 |

18 | 50201023123 × 139263586549 × 60466397701555612333765567 × C53313 |

19 | 775608719589345260583891023073879169 × C106685 |

20 | 352867 × 6210298470888313 × C213419 |

21 | 387347773 × 1620516511 × C426863 |

22 | 91798039513 × 7919244169465663354953966404923 × C853719 |

s_{0} to s_{9} are easy to factor. Ken Takusagawa made the
largest complete factorization by finding a 44-digit factor of s_{10}
with GMP-ECM.

Sean A. Irvine
found several large factors from s_{12} to s_{18} and added them to
Wikipedia. He has reported the following GMP-ECM work to me. s_{11}:
4590 curves at B1=11e6. s_{12}: 1000 curves at 1e6. s_{13}: 100 at 50k. s_{14}: 100 at 10k. s_{15}: 100 at 10k. s_{16}: 100 at 10k. Only few curves
above s_{16}.

Dario Alpern found the two known factors of s_{21} in 2006.

I trial
factored to 10^{12} in 2007 which gave the new factor 50201023123 in s_{18} (Irvine had
already found
a larger factor), and the first known factor of s_{22}.

Christophe Clavier found the factor 6210298470888313 of s_{20} in 2007, the factor 54062008753544850522999875710411 of s_{15} in
2009, the factor 60466397701555612333765567 of s_{18} in 2012, the
factor 775608719589345260583891023073879169 of s_{19} in 2012, the factor 115220560101116343072340969000241209 of s_{16} in 2019, and the factor 7919244169465663354953966404923 of s_{22} in 2019.

The smallest number without a known factor is s_{32}
which has around 874 million digits and unknown primality status.

The large composite divisors were proven composite with PrimeForm/GW. A base 3 Fermat prp test was performed with these results:

s18: 1080204337.....2994420807/(50201023123*139263586549*60466397701555612333765567) is composite: RES64: [439B8ADB88D2121B] (200.7818s+0.0867s) s19: 1166841411.....6400110443/775608719589345260583891023073879169 is composite: RES64: [A3CF26A0FD91EDAC] (848.6814s+25.3268s) s20: 1361518878.....6197545807/(352867*6210298470888313) is composite: [2FAA731D98F69934] (3957.6467s+0.4931s) s21: 1853733657.....3665735443/(387347773*1620516511) is composite: [1ED1B20FEACEAE0B] (17885.7170s+1.9189s) s22: 3436328472415493....354953966404923) is composite: RES64: [05CD01BEB95696E4] (15306.2212s+1.7304s)The below list shows known factors of s

In 1991 Ilan Vardi listed the prime factors below 5×10

Next, consider s

Dario Alpern computed the prime factors below 2×10

i Known prime factors of s_i --- -------------------------- 23 74587 24 206866598749591633 25 624116543851 26 27061 27 164299, 3229081, 16194353551, 7467340936147137520429 28 20929, 2621897371, 367660265779355658253 29 1171, 298483, 97562299 30 267160450861207 31 1679143 32 33 34 120823, 447841 35 2408563 36 38218903, 818864029 37 333457, 117602053 38 30241 39 4219, 1372512301 40 1085443, 156507607, 171818923 41 7603 42 1861, 84819199 43 44 23773, 290791 45 51769 46 1285540933 47 429547, 196973983 48 49 8323570543 50 51 179585227 52 53 9829, 15238744627 54 55 56 47101199947 57 763176709, 15517465963 58 59 2029, 38329, 320329, 3567469 60 61 50023, 1445860807 62 63 64 1459, 11234941 65 72091, 609421, 6377833117 66 67 245563, 3346633, 343870981 68 6367063, 10516273813 69 12763 70 384061 71 3799489, 8401963 72 73 11844787 74 35869, 26664046099 75 20161, 42428887 76 123427, 160200259 77 78 79 51973, 195502537 80 2437, 49848925897 81 82 6230971, 57741984199 83 84 24049307239 85 86 9436201 87 88 151477 89 41077, 2171593 90 91 92 15373, 21661, 1386013 93 429446209 94 12593197 95 535879 96 1407223, 1143074323 97 2748116731 98 20479 99 100 537037, 142997167 101 73381873 102 103 4519 104 13291 105 9123469 106 97549 107 1101733, 116289529 108 297589, 549733 109 118953139 110 943567 111 1285633 112 113 132637, 45352921 114 473611 115 6906223 116 148207, 51205591, 331215169 117 17977 118 119 54919, 644702497 120 121 122 123 1944721, 234341827 124 125 126 32779, 1604059579 127 128 129 130 26778949, 142670533 131 36680803 132 133 134 2541461581 135 256204723, 315917071 136 27031 137 8221, 149197, 186749617 138 769858669, 11304677191 139 46380739 140 6469 141 6886081 142 1182253, 2211019 143 144 615721 145 2564857, 3113936041 146 13147, 8186719 147 148 7205893 149 150 151 2686393, 369582877 152 153 154 155 156 232051, 438049093 157 158 159 160 161 64621, 1530229, 8244391 162 163 164 165 166 1763911, 6051648259 167 168 169 695131, 3503498041 170 215899 171 172 35099257 173 839437, 792599053 174 175 46022962369 176 31159, 28384141 177 178 4508756437 179 180 57853 181 182 183 107053 184 60919, 505123 185 39360469 186 187 188 7948981 189 190 2986572571 191 192 193 12436189 194 80173, 11073379, 734468401 195 1156390513, 61625866987 196 197 562909, 1914031477 198 13163791, 24518173 199 19597, 150649, 95037412693 200 765241

sylvester-factors.txt contains all known prime
factors, written as [i, p] where p divides s_{i}, and sorted by p.

The factors include:
All factors below 2^{32} for * all* i (this page stops at i
= 200),
all factors below 10^{11} for i ≤ 200,
all factors below 10^{12} for i ≤ 25,
all factors of any size for i ≤ 10,
and some factors above 10^{12} for 11 ≤ i ≤ 24.

Let p be a fixed prime. The recursive formula s_{i+1} = s_{i} × (s_{i}−1) +
1 means that the sequence (s_{i}) modulo p must eventually enter a cycle
since there is a finite number of possible values modulo p. Therefore, if a
cycle in (s_{i}) modulo p is detected before reaching a number divisible
by p, then p will never be a factor in Sylvester's sequence.

Vardi reported that 1166 out of the first three million primes
(meaning up to 49979687 with no further factors below 5×10^{7}) are
prime factors of some s_{i}. My triple-checked computation says 1167
factors (the first 1167 entries in sylvester-factors.txt), corresponding to 1 in
2571. My additional computations say that 8181 of the 203280221 primes below 2^{32}
are factors (1 in 24848).

Here is a PARI/GP
function to test whether p divides s_{i}, using modular multiplication
without computing s_{i}:

isfactor(i,p) = local(j,s);s=2%p;for(j=1,i,s=(s*(s-1)+1)%p);s==0It is slow and unsuited to search new factors but it can verify all known factors in 10 GHz minutes with this PARI/GP function:

verify_factors(file) = { local(f,c,j,p,n); f=readvec(file); c=j=p=0; for(n=1,#f, if(isfactor(f[n][1],f[n][2]), c++); if(isfactor(f[n][1],(f[n][2])^2), j++); if(isprime(f[n][2]), p++); if (n>1 && f[n][2]<=f[n-1][2], print("Number "n" not increasing")); ); print(#f" tested, "p" primes, "c" correct factors, "#f-c" incorrect, "j" square factors."); } ? verify_factors("sylvester-factors.txt") 8228 tested, 8228 primes, 8228 correct factors, 0 incorrect, 0 square factors.

**Links**

Wikipedia, Sylvester's sequence.

MathWorld by Eric W. Weisstein,
Sylvester's Sequence.

Ken's blog by Ken Takusagawa,
Factoring Sylvester's sequence and
Sylvester
10th factored.

Mersenneforum.org, Number 100
("alpertron" is Dario Alpern) and Sylvester's
Sequence.

Reference copied from MathWorld (without having access to it):

Vardi, I. "Are All Euclid Numbers Squarefree?" and "`PowerMod` to the Rescue."
§5.1 and 5.2 in
Computational Recreations in Mathematica. Reading, MA: Addison-Wesley, pp. 82-89, 1991.

This page is at http://primerecords.dk/sylvester-factors.htm
and licensed under the GFDL.

Page by Jens Kruse Andersen,
jens.k.a@get2net.dk home

First uploaded 27 August 2007. Last updated 5 September 2020.