Problems & Puzzles: Puzzles

Puzzle 325. Zeisel numbers

J. K. Andersen proposes the following puzzle:

A Zeisel Number, Z(n), is a number with distinct prime factors p_1*p_2*...*p_n, where n>=3, p_0=1, p_i = A*p_(i-1)+B for i = 1..n and constant integers A, B.

See the following article.

 
n Smallest n-Zeisel number, Z(n)  A, B
3 105 = 3*5*7 1,2
4 114985 = 5*13*29*61 2,3
5 1136972771 = 11*31*71*151*311 2,9
6 717429818501= 11*31*71*151*311*631 2,9
7 ?  
8 ?  
9 ?  
10 ?  

This Curio (ignore silly emirp claim) has a pretty case (the smallest?) with 7 prime factors:

31*331*3331*33331*333331*3333331*33333331, (A,B)=(10,21)

Questions (by Andersen):

1. How large do you expect to be the smallest n-Zeisel Number, Z(n)?

2. Find the smallest n-Zeisel Number Z(n), for n = 7, 8, 9, 10, ...

3. Find a titanic n-Zeisel Number with n prime factors for n = 3, 4, 5, ...

4. Find a Zeisel Number with at least 10000 digits. How large can you find?

After I received the Andersen's suggestion for this puzzle I made a code in Ubasic, in order to verify the data sent by him; as a by-product of the confirmation, my own contributions to the Andersen's questions are following statements as answers to question 2:

a) A & B for Z(7) are: 10, 21
b) A & B for Z(8) are:   2, 24089
c) A & B for Z(9) are:   3,232820
d) A & B for Z(10) are: 4,124767


Karsten Meyer wrote (Set. 2005):
 

"42193497392022209194699696424911 =
31*331*3331*33331*333331*3333331*33333331" is not the only and not the smallest curiousity in Zeisel numbers. 823055661241 = 11*211*4211*84211 with
(a,b)=(20,-9) is a curiousity too, because every following prime number inherit its prime number before. 717429818501 = 11*31*71*151*311*631 is also curious, because it is an existing Sequence: 1, 3, 7, 15, 31, 63 is 2^n-1.

***

Anton Vrba wrote (December, 2005):

Further to Andersen's question/challange No 3
 
A) if A = 2^607-169662 and B=169661 then
p_1, is prime (Mersenne) p_2 and p_3 are  PRP . Jens Kruse Anderson quickly prooved them prime using Marcel Martin's Primo
 
Hence p_1*p_2*p_3  is a 1097  Zeisel number
 

B) if  A=10^100 - 5695475 and B=5695742 then

    p_1, p_2, p_3 and p_4 are prime (Marcel Martin's Primo)
 
    Hence p_1*p_2*p_3*p_4 is a 1000 digit  Zeisel number

C) if A= 129285*157#-784616 and B=784617 then
    p_1,p_2, p_3, p_4 and p_5 are prime .
    Hence p_1*p_2*p_3*p_4*p_5 is a 1000 digit Zeisel number
 
And another interesting result for a Zeisel number just greater than 10^999 (1000 digits  
 
D) if A=IntegerPart[Sqrt[10^333]]+158 and B=488 then
     p_1,p_2 and p_3 are prime and the 1000 digit Zeisal number
     p_1*p_2*p_3 = 10...(161 zeros)...07612..etc...049 ie. total 163 zeros after the leading digit 'one'.
 
E) Finding titanic Zeisel numbers is easy, but there is only ONE "smallest titanic Zeisel number" and it is given by:
 
A=
31622776601683793319988935444327185337195551393252\
16826857504852792594438639238221332795089011104326\
54043121391518782908565638947316526248167789448817\
47194186169170037
 
and
B=
22906038736391937293832140475304452125784933114577\
2350660252502170508689694560093070
 
p_1,p_2 and p_3 are all prime (Marcel Martin Primo)
 
p_1*p_2*p_3 evaluates to
 
10000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000000000000\
00000000000000000000000000000000000000000008130662\
74897977182250381064075903168304830703002675756704\
03248927014950715734861755019610170000000000000000\
00000000000000000000000000000000000000000000000000\
00000000000000000724352545790437558095806875926741\
48407235596476734062150866991414176380308653759388\
03898075437542199453639887609155197303127618319533\
22215465083036898894117934215969523853489545270767\
46217651660467266510957803645143024829618817905868\
78319205632216507757024832696509356482986582753144\
85411649912026369110743977107044303482079216895528\
13068949720229758639748767830027620066845392018745\
86948328793378480497532051865535092373925989087063\
16732038921520068210628386993936605840494332424247\
45478915218092585642601546888915942284849036042605\
18933491983660422865952407558689292928551028592729
 

Sorry it might not be the smallest, one needs to search (1,X333) which might yield the smallest Zeisel. I calculated that there are about 1.9*10^87 p_1 primes before the below number is exceeded , so the chances are good.

 
Any way, the corrected statement is that the Zeisel below is the smallest titanic Zeisel with A and B > 1

 

***

 

 


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