Problems & Puzzles: Puzzles

Puzzle 629. Non-primitive Carmichael numbers

Let's suppose that we define a Perfect Non-Primitive Carmichael number (PNPC) any Carmichael number such that it can be "factorized" completely in terms of k>1 smaller Carmichael numbers:

C = C1*C2*..Ck = PNPC

While we define a Simple Non-Primitive Carmichael number (SNPC) any Carmichael number such that it is not a PNPC but it can be "factorized" partially as a product of at least one Carmichael numbers and a Cofactor F>1 that is not a PNPC:

C = (C1*C2*...Ck)*F = SNPC

A Carmichael number C that is neither a PNPC nor a SNPC is defined as a Primitive Carmichael Number (PC)

The smallest examples of each one are:

Smallest PNPC =  509033161 = 1729*294409
Smallest SNPC = 63973 = 1729*37
Smallest PC = 561

BTW, the smallest PNPC is given by the young Renaud Lifchitz in one of his papers, from 2002.

Q1. Develop an efficient algorithm to determine if a given Carmichael number C is PNPC or SNPC or PC.

Q2. Using the Yoshio Mimura table of Carmichael Numbers up to 10^12, classify each entry as PNPC or SNPC or PC and fill the following Table:

 Qty of factors Total Carmichael numbers up to 10^12 PNPC SNPC PC 3 1000 0 0 1000 4 2102 5 3156 6 1714 7 262 8 7 Totals 8241

Another issue related to a PNPC Carmichael number C is to determine if its (perfect) factorization is unique or not, disregarding the factors order.

As a matter of fact, Emmanuel Vantieghem studied the smallest Carmichael numbers that are the (perfect) product of two Carmichael numbers. See his very recent A207041 at the OEIS.

Incidentally he found that 84127131361 = 7*13*31*37*61*73*181 -the 9th member of his sequence- accepts two ways of (perfect) factorization: 84127131361 = 15841 * 5310721 = 172081 * 488881, where all the integers are Carmichaels, of course.

Nevertheless, for sure it must exist Carmichaels integers accepting W>2 distinct ways of (perfect) factorizations in terms of smaller Carmichael factors.

Q3. Send your best Carmichael having the largest quantity of distinct Carmichael (perfect) factorizations, W.

Contributions came from Fred Schneider, Emmanuel Vantieghem & Hakan Summakoğlu

***

Fred wrote:

1) You can't have a PNPC of < 6 factors.

2) Using Mr. Lifchitz's method, any Carmichael constructed as a product of 3 or more Carmichaels would have a non-unique "factorization" (NUF).

c1 = 7207201
c2 = 230630401
c3 = 56951294401

Example = C = (c1*c2) * c3 =  c1 * (c2*c3)

One thing that would be really interesting to me is a NUF-PNPC which is non-trivial

Consider : C = x1x2 = y1y2 where C, x1, x2, y1, y2 are all Carmichael numbers
and the gcd between any x(i) and y(j) is not a Carmichael.

Let's call C strongly non-trivial if C can't be expressed as the product of three or more Carmichael numbers.

I wonder if any such numbers exist.  The effect would be have to prime factors of the Carmichael divisors interleaved.

***

Emmanuel wrote:

Q1 : The situation is very complicated.  If one denotes by  Sn  the set of all Carmichael numbers that can be written as a product of  n  Carmichael numbers, then these sets form a very chaotic structure for the relation of inclusion.  I illustrate this with some examples :

The first ones of  S2  are:

509033161, 1836304561, 5394826801, 20064165121, 25594002721, 47782272385, 59970791881,
75527369281, 84127131361, 96578912521, 116087568961, 278585544601, .... (this is A207041 at the OEIS)

The first ones of S3  are : 20891229960601, 130336910332993, 1858266723268501, 2094319836529921, 3452661598186801, 3963695148774361, 4137856167537001, 5665575729414601, 34047477574249321, 39877189446732001, 60524817082337881, ...

(I submitted the first one at PrimeCurios! but it still awaits for approval).

And here are the first ones of S4  starts like that : 521649490603448721200401, 2022869722846930429318561, 3349825341543271940706721, 6087026898915845017172401, 19677515205436601941185121, 25429332308691672225431281, 39431226379497756087285001, ...

(I know many elements of  S5  but off the first one, say  a, I only know :

g1 < a < g2

with  g1 = 2642016264494523233035163069625  (31 digits)
g2 = 187368552845933395494213660170881  (33 digits)).

There are elements in  S2  that are not in  S3 (509033161 of course)
There are elements in  S3  that are not in  S2 (1858266723268501)
There are elements in  S3  that are also in  S2 (20891229960601)
There are elements in  S4  that are in  S3  and  S2  (19677515205436601941185121)
There are elements in  S4  that are in  S3  but not in  S2 ( 521649490603448721200401)
There are elements in  S4  that are in  S2  but not in  S3  ( 3349825341543271940706721)
There are elements in  S4  that are not in  S2  and not in  S3 (6087026898915845017172401)
All of this indicates that the question of finding all decompositions will become more and more complicated ...

Q2, these are the results (I omitted the header) :

3   1000   0     0    1000
4   2102   0    103  1999
5   3156   0    265  2891
6   1714   5    253  1456

7    262   10    90    162
8     7       0     6       1
t   8241  15  717  7509

I made a more extensive table for all Carmichael numbers < 10^8 :

3    35586       0          0         35586
4    40685       0       1361      39324
5   178063      0       4371     173692
6   414660     79     13622    400959
7   460553    373    29456    430724
8   223997    735    35983    187279
9    44993     640    17339     27014
10    3058      175     2109        774
11      49          8          39          2
tt   1401644  2010  104280   1295354

A little bit surprising is the fact that about  92%  of the Carmichael numbers are PC (and that % seems to grow slightly.  Does it becomes 100 ?).

Q3 : my 'champion' for multiple representations is : 767120409871716014168927041 :

767120409871716014168927041 = 488881*552721*3057601*928482241
= 488881*3057601*10402561*49333201
= 488881*552721*2838928228563841
= 488881*10402561*150841244710801
= 488881*49333201*31806880916161
= 488881*140241361*11188819320001
= 488881*928482241*1690000282321
= 488881*12517265041*125357675521
= 552721*12517265041*110878699681
= 49333201*140241361*110878699681
= 10402561*73743418555461103681
= 110878699681*6918555250726561
= 1041928957441*736250206305601
= 2825226381601*271525289041441
= 17193076256161*44617984498081
(15 representations).

***

Hakan reported

For Q2 exactly the same table that reported above Emmanuel.

Q3: Smallest Carmichael Number For W=3:

65121765643441 = 13 * 17 * 37 * 41 * 61 * 73 * 181 * 241 =
115921 * 561777121 = 488881 * 133205761 = 5310721 * 12262321

***

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