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RE: [PrimeNumbers] A 155-digit brilliant number

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  • Hadley, Thomas H (Tom), ALABS
    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
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      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.

      Tom Hadley

      -----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
      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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