Discussion between Enzo Creti and Peter

5:26 AM

2 9996044114449441 16
13 9996429851232989 16
14 9996232805624411 16
15 9996363229838603 16
16 9996178946079187 16
17 9996429851232989 16
18 9996961813848743 16
19 9996280902490237 16
20 9996235714451341 16
21 9996204119753843 16
7699965577

@Peter

Enzo Creti

7:09 AM

@Peter in the ec prime project i jumped from 430k to 470k...i believe that there is a huge gap at least 100k between 366770 and the next esponent leading to a prime

it seems that ec primes pop up with a sort of periodicity...but it is too difficult for me to find a pattern

Peter

7:26 AM

@EnzoCreti Nice example, let us try "l=10^9"

That there are some surprising occurences in the exponents of the ec-primes does not mean that there is a pattern. If you want, you can check the exponents with ending digits 1456. We can call this new project the 1456-project.

FF 48 (show) 815915283247897734345611269596115894264300034423<48> = 8519215538587009<16> · 9580989104325229<16> · 9996204119753843<16>

40!-7699965577

Enzo Creti

at home i will try

Peter

7:49 AM

Astonishing : I sieved out the prime factors upto $10^6$ of the exponents with ending digits 1456 upto $10^7$. Out of the 1000 candidates, 234 are still remaining.

Enzo Creti

@Peter ec primes are not random at all...but yet I did not find somebody who could explain that...on Mersenne forum they say mine is numerology not math...I don't know...if a pattern exists, it's a job for a top mathematician i think...

we are not anymore in the age of Fibonacci, when simple patterns were found in a easy way...

moreover Fibonacci stolen his sequence from indians

Enzo Creti

8:14 AM

I have never understod why they call it Fibonacci sequence when the discoverer was an indian

Peter

9:04 AM

Probably noone can be sure concerning historical facts if lying is apparently normal. Who knows whether Pythagoras found $a^2+b^2=c^2$ or someone else ? The example of Albert Einstein shows that "fake-news" are common, unfortunately.

Even worse, human beings seem to be even proud of faking history.

Enzo Creti

the famous de l'Hopital theorem was not discovered by the mediocre mathematician maquis de l'hopital but was discovered by the glorious Bernoulli...de l'Hopital just payed Bernoulli for having mathematical discoveries

Peter

199 candidates "survive" the factors upto $10^7$ :

[51456, 141456, 151456, 271456, 281456, 291456, 391456, 401456, 441456, 491456,
531456, 541456, 631456, 701456, 811456, 841456, 861456, 871456, 901456, 961456,
1001456, 1161456, 1181456, 1191456, 1241456, 1261456, 1301456, 1381456, 1481456,
1521456, 1551456, 1571456, 1581456, 1601456, 1631456, 1641456, 1761456, 1771456
, 1821456, 1891456, 1941456, 1951456, 2001456, 2081456, 2181456, 2261456, 228145
6, 2331456, 2341456, 2431456, 2591456, 2611456, 2641456, 2651456, 2691456, 27014
56, 2791456, 2881456, 2901456, 2981456, 3001456, 3041456, 3061456, 3151456, 3171

Enzo Creti

@Peter 51456 and 541456 have also the astounding property that their difference is a perfect square!

Peter

@EnzoCreti As said, who wants to really know all this ?

Enzo Creti

i think nobody

Peter

9:14 AM

Maybe, someone hated L'Hospital and invented this story to destroy his fame.

Enzo Creti

@Peter i don't think, they say that letters have been found which proof that the theorem was discovered by Bernoulli

Peter

A really sad habit , the faking of history , anyway.

Enzo Creti

yes

Peter

Einstein is the worst example , consider for example the Hilbert-Einstein-case.

Enzo Creti

why?

Peter

9:20 AM

Einstein was a plagiarist. Many ideas were not from him , and many of them are even useless.

Enzo Creti

Among the candidates are there some candidates C such that C-541456 is a perfect square?

or such that C-51456 is a perfect square?

if one of them exist, probably it is an ec prime!

Einsten special relativity are essentially Lorentz transformations...most of the mathematial background of general relativity is due to Riemann

@Peter

Peter

The whole relativity theory was an unlucky "discovery" , to say the least.

Enzo Creti

M-theory is probably even worst

Peter

What does the "M" stand for ?

Enzo Creti

9:36 AM

good question! I saw a video on youtube...mother of all theories...mistery theory...M reversed is the W of Witten the discoverer...membrane...other say M stands for an english term I dont remember but I think it means something like mist, fog...that to say the complete confusion around this "magic" theory!

@peter

D-branes, do you really think they exist?

Peter

Also annoying are the ultrafinitists denying the concept of induction and the existence of huge numbers like Graham's number. D-branes ? Already the name makes me sceptical.

Enzo Creti

yes according to the theory, we humans are like flies trapped on rusks which are the D-branes

@Peter

we are glued to the D-branes

Peter

They should consume less drugs, my theory.

Enzo Creti

at least Galileo had this in mind: what cannot be proved, it is not science!

it is fantascience

Peter

In some sense, relativists are ultrafinitists, the speed of light replaces an infinite speed. Popper said : A statement is only scientific , if it can be DISPROVEN. A statement in physics can never be PROVEN, we can only say that it coincides so well with our experiences that there is almost no doubt that it is true.

Enzo Creti

9:49 AM

@Peter is there a candidate C such that C-51456 or C-541456 is a perfect square?

Peter

The sadest aspect is probably that time travelling actually seems to be taken serious. This is not science, not even science fiction, it is religion.

Enzo Creti

yes...maybe angels can travel in time...humans i think will never

Peter

I continued the sieving upto $2\cdot 10^7$. The square-vectors are :

? print(select(m->issquare(m-51456)==1,v))
[51456, 141456, 541456, 861456, 1261456, 2611456, 4051456, 7891456, 8461456, 905
1456, 9661456]
? print(select(m->issquare(m-541456)==1,v))
[541456, 631456, 701456, 901456, 1181456, 2791456, 3431456, 3781456, 6791456, 78
31456]
? length(v)
%25 = 188
?

Enzo Creti

i will try with sage ec(861456)?

maybe it is prp

Peter

Expressions for factordb :

? for(j=1,length(v),print(text(v[j])))
(2^51456-1)*10^15490+2^51455-1
(2^141456-1)*10^42583+2^141455-1
(2^151456-1)*10^45593+2^151455-1
(2^271456-1)*10^81717+2^271455-1
(2^281456-1)*10^84727+2^281455-1
(2^291456-1)*10^87737+2^291455-1
(2^391456-1)*10^117840+2^391455-1
(2^401456-1)*10^120850+2^401455-1
(2^441456-1)*10^132892+2^441455-1
(2^491456-1)*10^147943+2^491455-1
(2^531456-1)*10^159984+2^531455-1
(2^541456-1)*10^162995+2^541455-1
(2^631456-1)*10^190087+2^631455-1
(2^701456-1)*10^211159+2^701455-1

Enzo Creti

9:57 AM

i rule out all with exponents<430k

are there perfect square among them which are of the form 7^2*k^2?

i mean C-51456 or C-541456 is a perfect square 7^2*k^2?

Peter

Currently, I continue the sieving upto $3\cdot 10^7$

Enzo Creti

ok

Peter

Yes, $7891456=7^2\cdot 400^2+51456$

Enzo Creti

it is prp

i guess

Peter

Curious that "789" and "456" form a digit - arithmetic-progression

Enzo Creti

10:08 AM

yes

how to test if ec(7891456) is prp?

Peter

This number is huge, I think we have to find a factor.

Enzo Creti

which is the expression?

(2^7891456-1)*10^2375565+2^7891455-1

is that right?

Peter

(2^7891456-1)*10^2375565+2^7891455-1

Enzo Creti

@Peter

Peter

Yes, I entered in factordb, but it seems to hang

Enzo Creti

10:13 AM

entered in Sage i think it will took a month!

i will also ask on mersenne forum

@Peter

Peter

4 751 130 digits

U 4751130 (show) 10^2375565*(2^7891456-1)+2^7891455-1<4751130> = 9235544354...67<4751130>

Enzo Creti

how can we look for a factor?

Peter

With my special function, I currently run trial division.

Enzo Creti

ok

Peter

passed 200M

Enzo Creti

10:24 AM

what does it mean?

@Peter

Peter

no factor below 200 millions

passed 400M

Enzo Creti

if it is actually a prp, i will print it and i will hang it in my bedroom

chances to be prime anyway are extremely small!

@Peter

Peter

10:48 AM

passed $2\cdot 10^9$

Enzo Creti

11:07 AM

the chances it is prime?

@Peter

Peter

11:21 AM

passed $4\cdot 10^9$

Enzo Creti

@Peter i started also with Pfgw...I calculated about 10 hours for an answer!

Peter

Chances about 1 : 277 812

Enzo Creti

Maybe i was wrong it seems that Pfgw is a continental drift

so we have to find a factor

Peter

4 million digits still is a challange !

I could program p-1-method, but it will be very slow indeed.

Enzo Creti

you could ask Robert Frost maybe?

Peter

11:56 AM

The following candidates, including 7891456 lead to a number with residue 6 mod 7 :

? print(select(m->f(m,7)==6,v))
[271456, 401456, 871456, 901456, 1001456, 1181456, 1571456, 1601456, 1891456, 22
81456, 2591456, 2701456, 2881456, 2981456, 3581456, 3871456, 3971456, 4571456, 5
671456, 5801456, 6791456, 6871456, 7291456, 7471456, 7891456, 7991456, 8671456,
8701456, 8771456, 9271456, 9301456, 9371456, 9691456, 9791456]
?

173 candidates left after sieving to 50 millions

? print(v)
[51456, 141456, 151456, 271456, 281456, 291456, 391456, 401456, 441456, 531456,
541456, 631456, 811456, 841456, 861456, 871456, 901456, 1001456, 1161456, 118145
6, 1191456, 1241456, 1261456, 1381456, 1481456, 1521456, 1571456, 1581456, 16014
56, 1631456, 1641456, 1761456, 1821456, 1891456, 1941456, 1951456, 2081456, 2181
456, 2261456, 2281456, 2331456, 2341456, 2431456, 2591456, 2611456, 2641456, 265
1456, 2701456, 2791456, 2881456, 2981456, 3041456, 3061456, 3171456, 3251456, 33
21456, 3411456, 3431456, 3581456, 3591456, 3601456, 3671456, 3701456, 3751456, 3

Enzo Creti

@Peter so if the ec number is prime then the main conjecture is disproven?

Peter

yes !

Enzo Creti

so now I think that the number is not probable prime

@Peter which of them do not lead to a residue 6 mod 7?

Enzo Creti

12:22 PM

@Peter with Pfgw I give up with ec(7891455) and I started ec(631456)

with Pfgw ec(7891455) takes more than 20 days!

Peter

12:39 PM

Above, I posted the remaining candidates, and the exponents leading to residue 6 mod 7.

Enzo Creti

ok

Peter

If we are not lucky, the smallest factor (if not out of reach) will be difficult to find.

Enzo Creti

yes

Peter

We can try trial division with SAGE

Enzo Creti

i don't know the command

Peter

12:55 PM

We only need gcd and powermod (or powmod)

for the p-1-method and only mod for trial division

Enzo Creti

ok

let's try powermod give me an input

@Peter

Peter

powermod(7,17,37)

Enzo Creti

1:11 PM

i wrote 7.powermod(17,37) and returned 16

@Peter

Peter

correct

Enzo Creti

ok

Peter

Let us try the following pseudocode, first for a small number

n=2^67-1

x=3

j=1

Enzo Creti

ok

Peter

while gcd(x-1,n) = 1

j=j+1

x=powermod(x,j,n)

end while

print gcd(x-1,n)

Enzo Creti

1:14 PM

no something doesnt work

x=x.powermod(j,n)

Peter

it is only a pseudocode, you have to translate it

output should be 193707721

Enzo Creti

i understand that we have to use while true:

Peter

Isn't the syntax similar to that in python ?

Enzo Creti

yes but not exactly

Peter

Do you have a description how the commands work ?

Enzo Creti

1:29 PM

no

there is also indentation in Python and Sage

while gcd(x-1,n)==1:
....: j=j+1
....: x=x.powermod(j,n)
....: print gcd(x-1,n)

Peter

"end while" missing

Enzo Creti

Now the program works but in the output a series of 1 came out and at the end 193707721

@Peter

Peter

Somehow we have to close the while-loop

Enzo Creti

@Peter the final result is correct but I don't understand why all that 1's before...while loops perhaps closes with indentation

ec(631456) is composite

Peter

OK

If we cannot close the while-loop we have to accept the unnecessary ones (the program then outputs gcd(x-1,n) every time which is of course 1 until a factor is found)

Enzo Creti

1:41 PM

age: while gcd(x-1,n)==1:
....: j=j+1
....: x=x.powermod(j,n)
....: break
....: print gcd(x-1,n)
....:
193707721

@Peter now correct?

Peter

OK, great!

To display the progress, add "print j" after "j=j+1"

Enzo Creti

ok

ok now?

@Peter

Last outputs ?

193707721 the same!

And the numbers j ?

1:45 PM

no

Peter

The second last output should be 2677

Enzo Creti

Why a second output?

Peter

The program should display the number j until the factor is found.

Enzo Creti

immediatly the program found the output

Peter

Otherwise, we do not see where it arrived.

Enzo Creti

1:49 PM

@Peter

Peter

The final number is the whole output ?

Enzo Creti

it seems so

Peter

How did you add the print-command ?

Enzo Creti

j=j+1; print j

Peter

Better use another line. Maybe we need println or something like that instead.

Enzo Creti

1:53 PM

@Peter the routine shouldn't finish with 193707721?

Peter

Yes, but it should display 1 2 3 ... 2677 before

Sorry, start is 2

Enzo Creti

age: while gcd(x-1,n)==1:
....: j=j+1
....: x=x.powermod(j,n)
....: break
....: print j
....: print gcd(x-1,n)
....:
2677
193707721

@Peter now correct?

Peter

Yes, correct.

A moment, the othere numbers were not displayed ?

Enzo Creti

no why?

Peter

"print j" before "break" , otherwise we have no clue later where the program arrived.

Enzo Creti

2:01 PM

age: while gcd(x-1,n)==1:
....: j=j+1
....: x=x.powermod(j,n)
....: print j
....: break
....: print gcd(x-1,n)
....:
193707721
sage

why it doesnt work?

@Peter

Peter

No idea

Maybe println ?

Enzo Creti

no

Peter

is "break" actually the command to close a while-loop ?

The Syntax seems to be even worse than that in Python.

We could run the large number, but it would be a "blind flight"

Maybe, you do that and try to find a solution in another window.

@EnzoCreti

Enzo Creti

2:20 PM

sage: while gcd(x-1,n)==1:
....: j=j+1
....: x=x.powermod(j,n)
....: print j; print gcd(x-1,n)
....:
2678
193707721
sage:

@Peter

Peter

the first number is off by one and still the progress is not displayed.

OK, let us start the program without the display ...

Enzo Creti

@Peter No factors till 32*10^9. Quitting my attempt. on mersenne forum

Peter

Further than my search, maybe p-1-method is successful.

Enzo Creti

2:43 PM

@Peter

sage: while true:
....: n=2^67-1
....: x=3
....: j=1
....: while true:
....: gcd(x-1,n)==1
....: j=j+1
....: x=x.powermod(j,n)
....: print j
....: pass
....: print gcd(x-1,n)

Peter

And this works as desired ?

Enzo Creti

now the program is displaying a lot of numbers with false

36781 false

...

Peter

Strange, what does "36781" mean ?

Enzo Creti

@Peter are all the j's for which the condition is not true?

Peter

This number should not occur at all.

OK, see you later.

SAGE might be fast, but it is not user-friendly at all.

Enzo Creti

2:50 PM

no

@Peter sage: n=2^67-1
sage: x=3
sage: j=1
sage: while gcd(x-1,n)==1:
....: j=j+1
....: print j
....: x=x.powermod(j,n)
....: pass
....: print gcd(x-1,n)
....:
2
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Now seems to work

@Peter

2672
2673
2674
2675
2676
2677
193707721

see you tomorrow for Sage

Enzo Creti

4:12 PM

@Peter at home have you some routine for Pari?

shall we search for a factor of the huge ec >32*10^9?

or shall i use Yafu?

or have you some project?

Peter

5:54 PM