## RE: [PrimeNumbers] A 155-digit brilliant number

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• Paul, I don t think 77 is brilliant radix 10. The factors don t have the same number of digits base 10. They do for base 5, or 12 for example, but not 10.
Message 1 of 2 , Mar 25, 2003
Paul,
I don't think 77 is brilliant radix 10. The factors don't have the same number of digits base 10. They do for base 5, or 12 for example, but not 10.

Nit-picking aside, thanks for the info. :-)

I noticed your two factors were within a factor of 2 of each other and checked to see if they were 2-brilliant radix 2, and sure enough, they are.
The smaller number is 1000001110001100001101011101100010101010000110000110010001100000011000\
1111111101101111011001100011001110100001110110010000111111111100000110\
1010110100101000100100101011001110100001010101110010100000001110111100\
11100111110010000100110000111011001100100011011

and the larger is
1111100111100011111010000000111111110110000011111100111100001001111110\
0001110100010001110011011000000110110110001111010101001001110111111000\
1000100110111000001001110100001011000010101111100000000011110001001100\
01000101111011110110001101101001011011110011001

both with 257 bits.

-----Original Message-----
From: Paul Leyland [mailto:pleyland@...]
Sent: Tuesday, March 25, 2003 7:28 AM
To: Prime Numbers
Subject: [PrimeNumbers] A 155-digit brilliant number

The definition of a brilliant number is one which has all its prime factors of the same length in some radix. Thus 9 and 77 are both bi-prime brilliant numbers radix 10 and 105 is tri-prime brilliant radix 10.

I came across a 155-digit biprime brilliant number radix ten quite by accident this morning. The number Cullen(529), or 529*2^529+1 was known to have the factorization 3*353*32633*c155 where c155 is a composite number with 155 decimal digits. Fifty days of SNFS sieving on an elderly laptop and a couple more days on a rather more modern machine yielded the result:

Factorization completed after 1129.49 seconds, at Tue Mar 25 08:16:00 2003
26901123434101975133292947542161047085252544228652110928854201837163768280596412
010176456552650994032013318683149975423269498628012318609211083655340805667
Probable prime factor 1 has 78 digits:
226057150432806012914295170850489320912553947427734205942045166863088151213977
Probable prime factor 2 has 78 digits:
119001426774589709194645348
510592872415307609390458520578011383635471325042971

Brilliant numbers are fairly uncommon (though easy to create) and it's nice to come across one by accident.

Paul

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