|
|
A329417
|
|
Carmichael numbers m that have at least 3 prime factors p such that (p-1)*p^2 divides m-p.
|
|
3
|
|
|
12876480001, 102293818705, 162303632569, 639554081761, 783962120161, 3224063844001, 4553777859841, 10276904735461, 40867660260505, 51496980091921, 51641004415105, 52412615611201, 52933062609505, 73892907966241, 97388953462801, 107862864807061, 182236335107905, 210587050134721
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
In 1950, Giuga conjectured that there are no composite numbers n for which 1^(n-1) + 2^(n-1) + ... + (n-1)^(n-1) == -1 (mod n). If such a number exists, then it must be a Carmichael number n such that (p-1)*p^2 divides n-p for every prime p dividing n.
|
|
REFERENCES
|
Giuseppe Giuga, Su una presumibile proprietà caratteristica dei numeri primi (in Italian), Istituto Lombardo Scienze e Lettere, Rendiconti di Classe di scienze matematiche e naturali, Vol. 83 (1950), pp. 511-528.
|
|
LINKS
|
|
|
EXAMPLE
|
m = 12876480001 is a term because it is a Carmichael number, and it has at least 3 prime factors p, {7, 11, 37}, such that (p-1)*p^2 divides m-p.
|
|
PROG
|
(Perl) use bigint; use ntheory ':all'; sub isok { my $m = $_[0]; is_carmichael($m) && (grep { ($m-$_) % (($_-1)*$_*$_) == 0 } factor($m)) >= 3 };
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|