Martin Hopf's prime numbers

Peter

00:11

Remark : Under the assumption that pqr is a Carmichael-number

Peter

16:34

@Aurel-BG I have two main projects. Do you want to join in ?

Aurel-BG

Hey Peter

what's the project ?

the ping scared me lol

Peter

First is to find large prime numbers of the form Martin Hopf wants to create huge Carmichael numbers (See above). Second is to find a positive integer N such that N,N^2,N^3 are all balanced (that is in its decimal expansion , all digits 0-9 occur equally often). For the bases b=2-9 , I found a solution , but none for base 10.

Aurel-BG

Yeah I can help for Carmichael Numbers

Peter

Third challenge : Find a prime number N as large as possible such that 9N is balanced.

Aurel-BG

oki

These projects are nice, I will try Carmichael numbers first

Can you give an exemple for the balanced numbers ?

Peter

16:51

A simple example is 1234567890 , each digit occurs exactly once. A balanced number (in base 10) must of course have 10k digits (k positiive integer) and must be divisible by 9. I found no other usefulc criterions.

Martin Hopf

17:07

Hello Peter and Aurel!

Aurel-BG

I see

Hey Martin !

Martin Hopf

Nice too see you both at the same time on my channel. :)

Aurel-BG

yay

Martin Hopf

BTW, it was Peter who created this room.

Peter

I failed to find a 5k or 6k digit prime number of the form you need

Martin Hopf

17:16

Aurel, feel free to ask if anything is unclear.

Aurel-BG

no problem

just checking currently some Carmichael numbers

Martin Hopf

@Peter no problem I have an 8k digit prime already. Never mind.

@Peter your described condition holds.

Peter

17:47

With your supercomputer , you will probably find my superbalanced number in some minutes. My computer is just too old.

Aurel-BG

why not, let's try :)

Peter

17:59

I tried it for several hours per day with many different approaches , maybe there is no solution , but I think , there is one.

Martin Hopf

@Aurel-BG what do you think of the new layout of mersenneforum.org ?

Aurel-BG

I don't know, I think it can be improved but I like it

Peter do you have some code for this one ?

and you Martin ?

Peter

Wait , I search my file

Aurel-BG

oki

Peter

bal(n,b)={h=0;v=digits(n,b);l=length(v);if(Mod(l,b)==0,d=l/b;z=[];for(a=0,b-1,for(c=1,d,z=concat(z,a)));if(vecsort(v)==z,h=1));h}

Aurel-BG

18:03

thx

Peter

this is the main-function , testing whether some number n is balanced in base b

A moment , this is online a function !

y1=[];for(a=0,9,for(b=1,2,y1=concat(y1,a)))

This creates the needed vector for the enumeration

t=20!;s=1;while(bal(s^2,10)<>1,q=0;gef=0;while(gef==0,t=t-2^15;u=Vec(numtoperm(20,t));s=fromdigits(vector(20,j,y1[u[j]]));x=digits(s^2);if(sum(j=1,length(x),x[j]^3)==8100,y=digits(s^3);if(sum(j=1,length(y),y[j]^3)==12150,if(bal(s,10)==1,q=q+1;print(q);if(bal(s^3,10)==1,gef=1)))));print(s))

Aurel-BG

do you use PARI GP online ? Sometimes I use PARI GP from Sagecell or Online PARI/GP compiler

Peter

the routine. It makes sense to set some seed with setrand(...)

I tried it , but the version I am aware of is not really fast either.

Martin Hopf

Did you kwow the newest PARI/GP developement 2.16 and higher now supports the primorial-operator.

Aurel-BG

oh nice

Peter

18:08

This is in fact an improvement !

Aurel-BG

indeed !

Peter

@Aurel-BG Does it work ?

Aurel-BG

I'm not sure I have 1 2 3 4 5 printed

I don't know what does it means

lol

I stopped

I will retry it

This is interesting how some Carmichael Numbers are palindrome in some bases

Peter

18:27

@Aurel-BG This is unfortunate , but it is how it works. It should finally find a case where N and N^2 are balanced and it continues until a solution is found.

Aurel-BG

ok let's try it again, by the way, when a number if found, it's printed entierly ?

Martin Hopf

@Aurel-BG for Carmichael numbers see also this thread @ mersenneforum.

Aurel-BG

okay :]

Peter

balanced example

A semiexample is displayed , yes , but it can take quite long. But chances for a solution are relatively good since I used a restriction for the digits.

Only , if the program stops , we are really done.

Aurel-BG

okay so let's go

Peter

18:36

Did you find Carmichaelnumbers palindrome in many bases , say , 5 ?

Aurel-BG

I found in base 2, 4 and 12

and 10 ofc with 101101

Peter

Not bad ! I asked a question about palindromic Poulet numbers , if I remember right only one other is known.

Peter

18:57

Did the program display a semisolution ?

Aurel-BG

noooo :(

Peter

Isn't 1105 palindrome in bases 2,4,19,24,64,84 ?

Aurel-BG

possible, I need a IsPalindrome() function lol

Peter

use Vecrev(digits(n,b))==digits(n,b) to check whether n is palindrome in base b

Aurel-BG

oh cool

thx I will try it :)

Peter

19:08

334153 [21, 76, 184, 326, 389, 442, 459, 468, 476, 504, 546, 816]

Do you require at least 3 digits in base b ?

Aurel-BG

yes

Peter

gp > for(j=1,length(a),s=a[j];v=select(b->(Vecrev(digits(s,b))==digits(s,b))*(b^2<s),[2..1000]);if(length(v)>4,print(s," ",v)))
15841 [90, 96, 99, 110, 120]
29341 [43, 47, 76, 77, 163]
41041 [144, 152, 171, 180, 190]
46657 [6, 36, 162, 192, 216]
115921 [252, 276, 280, 315, 322, 336]
278545 [2, 4, 98, 107, 128]
314821 [12, 60, 107, 111, 171, 477, 495, 530, 540]
334153 [21, 76, 184, 326, 389, 442, 459, 468, 476, 504, 546]
340561 [66, 156, 175, 211, 430, 440, 473, 495, 516, 528]
399001 [5, 60, 456, 475, 500, 525, 532, 570, 600]

Still no output ?

Aurel-BG

not yet

Peter

Oh

Tough !

What is the number output last ?

Aurel-BG

19:24

116

Peter

quite high , if there is no output soon , there might be something wrong with the code.

Aurel-BG

:/

Peter

but it can also be that a number passing my restriction is very rare

Aurel-BG

Yeah

Peter

I asked a question about a useful restriction for balanced numbers , no answer yet.

and I found a near miss where N and N^2 are balanced and the distribution of N^3 is very balanced.

67371265 [2, 4, 431, 455, 512, 561]
67653433 [934]
67902031 [423]
68154001 [494, 651]
69331969 [669]
72108421 [940]
72286501 [993]
74165065 [802]
75151441 [456]
75765313 [526]
78091201 [478, 667]
78120001 [372]
79411201 [621, 684]
79624621 [852]
80282161 [516, 572]
82929001 [420, 518]
84311569 [523, 540]
84350561 [729, 861, 985]
87318001 [652]
90698401 [775]
93614521 [650]
96895441 [432]
99861985 [513, 523]
100427041 [378]
101649241 [567, 577]
102090781 [734]
104404861 [642]

The base 16-example is striking !

Aurel-BG

19:34

wow amazing

can you check base 256 ?

Peter

I checked all bases upto 1 000 and displayed every example with some base in this range. I think, the database goes upto 10^17.

1064311039918369 [306]
1879280313387841 [990]
1925795774992321 [254]
4631170376004481 [182]
8219279193281101 [178]
10174701738146401 [60]
10552947068664001 [429]
11258301692358001 [205]
15694232604573601 [778]
16572787241647681 [456]

Aurel-BG

thx

Peter

this is not the end yet. Still no output ?

Aurel-BG

I'm at 151

Peter

OK, try to replace "t=t-2^15" by "t=random([1,20!])" , then there should occur some output.

16572787241647681 [456]
24547515564615121 [495]
27670494011668201 [525]
32496538458394081 [112]
39540852873849601 [404]
57942466794656731 [594]

Aurel-BG

19:41

oki

Peter

57942466794656731 [594]
72931224600780001 [624]
101533868438058001 [666]

a "beast"-Carmichael !

Aurel-BG

lmao

Peter

finished the database. You have something to report !

Aurel-BG

at-top level error with "t=t-2^15"

wait

I redo

nope doesn't work

oops

wait

I'm silly

bruh, can you give me the new code pls

Peter

s=1;while(bal(s^2,10)<>1,q=0;gef=0;while(gef==0,t=random([1,20!]);u=Vec(numtoperm(20,t));s=fromdigits(vector(20,j,y1[u[j]]));x=digits(s^2);if(sum(j=1,length(x),x[j]^3)==8100,y=digits(s^3);if(sum(j=1,length(y),y[j]^3)==12150,if(bal(s,10)==1,q=q+1;print(q);if(bal(s^3,10)==1,gef=1)))));print(s))

Is "bal" and "y1" still valid ?

Aurel-BG

19:49

seems ok

"1" printed

now 2

etc

Peter

hope now we get a "semisolution"

Python is sure much faster , but how can we program it there ?

Aurel-BG

I don't know Python at all

Peter

I never used it , but many seem to use it.

Aurel-BG

yeah, this is the n°1 code for employment apparently

Peter

Is there a fast online PARI tool that can be freely used ?

Aurel-BG

19:55

tutorialspoint.com/execute_pari_online.php you mean like this ?

Peter

I do not know whether this is the tool I tried. Did you try it ?

Aurel-BG

yes, I use it when I need "two" PARI GP at the same time

Peter

seems it is not the one I used. Is it fast ?

Aurel-BG

Yes I find it fast honestly

I like it :]

Peter

It is only a compiler , right ? I tried to enter "factor(24!-1)". How do I start it ?

factordb

Aurel-BG

20:07

do print(factor(24!-1))

don't remove the quit() at the end

can be buggy after

Peter

OK. Seems to work.

print(primepi(10^12)) seems to be somewhat slow

now an output ?

Aurel-BG

I'm at 64

Peter

again quite high.

I quit "print(primepi(10^12))"

Aurel-BG

20:23

Yeah, this is a drawback of this compiler, when it's inactive for around 20 sec, compiler stops àt all

Peter

20 seconds is not much ...

Aurel-BG

yeah, so the tips here is to print something in the console every 20 sec

or less

Peter

and then it continues to run ?

Aurel-BG

117, still no output

Peter

20:45

I will try the primepi(10^12) again ...

factordb

Aurel-BG

nice !

so I will leave

still no output

Peter

OK, if you find a result , please output it here

Aurel-BG

no problem !

See ya !

Peter

OK, bye

Aurel-BG

21:26

@Peter 78941160385273596204 :)

Transcript for

$Mathematics$ Martin Hopf's prime numbers