RE: [PrimeNumbers] A 155-digit brilliant number
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\
and the larger is
both with 257 bits.
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
Original number had 155 digits:
Probable prime factor 1 has 78 digits:
Probable prime factor 2 has 78 digits:
Brilliant numbers are fairly uncommon (though easy to create) and it's nice to come across one by accident.
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