Product launch

[xilman]

You may find this product amusing:

196175124573517092034922493422470908957491913295260473203518969764669282857
*
509748624946200879944668835201017220741864841301635474800405920115982269129


Paul

[Greenbank]

Why?

[JHansen]

=10^149+21553 It's rare to find a number this well suited for SNFS with only two large prime factors.

[Greenbank]

I'd thought that the Brilliant Number search had gone beyond this range but I was mistaken. The highest they have is 10^115+12741.

http://www.alpertron.com.ar/BRILLIANT.HTM

[smh]

This makes a nice candidate for a record brilliant number.

After prp testing, traildivision and a bit of ECM, i have just over 200 numbers left. More ECM will remove quite a few of these, but still a lot of SNFS remains.

I haven't checked how long a single number takes with SNFS though.

[xilman]

Originally Posted by smh

This makes a nice candidate for a record brilliant number.

After prp testing, traildivision and a bit of ECM, i have just over 200 numbers left. More ECM will remove quite a few of these, but still a lot of SNFS remains.

I haven't checked how long a single number takes with SNFS though.

The attached file gives the smallest prime factor of all odd 150-digit numbers less than 10^149 + 21711.

You will see that it proves that 10^149+21553 is indeed the smallest 150-digit brilliant number.


Paul

[alpertron]

Excellent!!!

I will upload the data tonight.

How long has this task ran? Would it be easy to fill the holes in the table?

[xilman]

Originally Posted by alpertron

Excellent!!!

I will upload the data tonight.

How long has this task ran? Would it be easy to fill the holes in the table?

I forget off-hand when I started, but it was several years ago. I'll see whether I can dig out better information.

It took so long mostly because the SNFS sieving was performed almost entirely on some of my museum pieces. At most three machines were working at once --- a PPro 233, a PII-300 and a Sun Ultra10. Quite often only one or two of them were sieving. The filtering, linear algebra and square root phases were run on a 1.33GHz Athlon. Each SNFS took about two weeks and I had to run dozens of them. You'll note that 21553/2 factorizations had to be completed. OK, many of them were done by trial division but a substantial ECM effort was needed too.

The SNFS factorizations are easy to spot in the file I attached. They are the ones with (pxx) appended. The ECM factorizations are, essentially, those without a (pxx) and for which the factors are larger than 6 or 7 digits.

I suspect that it will take a long time, possibly several years, to fill the gaps in the table, unless a number of people make a concerted effort. When I started the record stood at around 100 digits, possibly a bit more. You can see for yourself how much progress has been made in that time and how much more is required to fill the gaps.


Paul

[smh]

I count 77 SNFS tests.

For my 112 digits i did 210 SNFS tests, for 113 digits 351 and for 116 digits 53

Of course all of them were a lot easier.


Do you know what is the chance the brilliant shows up within lets say the first 25K n for numbers this size?

You might have been unlucky and have to run upto several hunderd K n

[Jens K Andersen]

Congratulations on a great brilliant number.

Originally Posted by xilman

The attached file gives the smallest prime factor of all odd 150-digit numbers less than 10^149 + 21711.

I guess Paul's brill.txt lists the first factor which was found.
ECM will often not find the smallest factor first.
My sieve to 400G found a smaller factor in 49 attached numbers below the brilliant 10^149+21553.
The largest difference was for 10^149+18537 with smallest factor 1616453 and Paul's 562916230505561.

[xilman]

Originally Posted by Jens K Andersen

I guess Paul's brill.txt lists the first factor which was found.
ECM will often not find the smallest factor first.
My sieve to 400G found a smaller factor in 49 attached numbers below the brilliant 10^149+21553.
The largest difference was for 10^149+18537 with smallest factor 1616453 and Paul's 562916230505561.

Yes, you are quite right.

Nonetheless, the brill.txt file lists a prime factor of each number in the range necessary to establish that 10^149+21553 is the smallest 150-digit brilliant number.

Paul