396 Jean-Marie De Koninck

228

+ 1

• the ninth Fermat number, a 78 digit number; it is a composite number, whose

factorization was obtained in 1980 by Brent and Pollard:

228

+ 1 = 1238926361552897

·93461639715357977769163558199606896584051237541638188580280321.

398 075 086 424 064 937 397 125 500 550 386 491 199 064 362 342 526

708 406 385 189 575 946 388 957 261 768 583 317

• the smallest prime factor of the RSA-576 number, a 174 digit number which

no one could factor, until December 2003; this number was finally factored by

Jens Franke, who obtained that the number

1881988129206079638386972394616504398071635633794173827007633

564229888597152346654853190606065047430453173880113033967161

99692321205734031879550656996221305168759307650257059

is the product of the two prime numbers

398075086424064937397125500550386491199064362342526708406

385189575946388957261768583317

and

472772146107435302536223071973048224632914695302097116459

852171130520711256363590397527.

229

+ 1

• the tenth Fermat number, a 155 digit number: it is a composite number, and

its smallest prime factor (found by Western in 1903) is

2 424 833 = 37 ·

216

+ 1.

2521

− 1

• the 13th Mersenne prime, a 157 digit number.

2607 − 1

• the

14th

Mersenne prime, a 183 digit number.

2210

+ 1

• the 11th Fermat number, a 309 digit number: it is a composite number, whose

smallest prime factor, namely

45 592 577 = 11 131 ·

212

+ 1,