Mathematical_Recreations-Kraitchik-2e

Numerical Pastimes 71
Two or more prime factors are known (but not the
complete factorisation) for n=151, 163, 173, 179, 18],
223,233,239,251.
n Factors of ~"- 1 - n Factors of ~"- 1
151 18 1~1 ·55 871 . 165 799· ~~3 18~ 187·196 687· 1 466
~ 33!t 951 449·!t 916 841
163 150 !t87· 704 161 ~88 1 899· 185 607· 6~!t 577
173 780758·1505447 !t89 479·1913·5787·176883
179 859·1488 ~51 508·54 !t17
181 48 441· 1 164 193
Only one prime factor is known for n= 131, 167, 191,
197,211,229.
n Factor of !t" - 1 n Factor of !t" - 1
131 !t63 197 7487
167 ~ 849 O!tS ~11 15198
191 888 ~!t9 1504073
2 n - 1 is composite but its factors are not known when
n=101, 103, 109, 137, 139, 149, 157, 193, 199,227,241,
257.
In addition some information has been gained about numbers
of the form 2" - 1 for values of n beyond Mersenne's
range. Thus, a factor is known for each of the following valuesofn:
263,281,283,317,337,359,367,397,419,431,443,
461,463,487,491,499,547,557,571,577,593,601,617,619,
641,659,683,719,743,761,827,829,839,857,877,881,883,
911, 929, 937, 941, 967.
For 2 397 -
1 no fewer than five divisors are known, namely
2,383, 6,353, 50,023, 53,993, 202,471, but the complete factorization
is not known.
A perfect number is a number equal to the sum of all of its
divisors (not including itself); for example, 6 = 1 + 2 + 3