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RE: [PrimeNumbers] AP with 385256-digit tau

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  • Jon Perry
    ... Big enough tau no doubt. But a big enough k? Jon Perry perry@globalnet.co.uk http://www.users.globalnet.co.uk/~perry
    Message 1 of 3 , Jan 29, 2002
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      >I reckon this is big enough, Jon?

      Big enough tau no doubt. But a big enough k?

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry
      http://www.users.globalnet.co.uk/~perry/maths
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      -----Original Message-----
      From: djbroadhurst [mailto:d.broadhurst@...]
      Sent: 29 January 2002 10:44
      To: primenumbers@yahoogroups.com
      Subject: [PrimeNumbers] AP with 385256-digit tau


      Primitive arithmetic progression with 385256-digit tau

      Let tau(n) be the number of divisors of n.
      Let p[k] be the k'th prime that is 1 mod 4.
      Let a[k] be the k'th member of the sequence with
      a[k+1] = a[k]*(a[k]+3) and a[1] = 1
      which grows like
      a[k] = O(c^(2^k)) with
      c = 1.52652645702134522042544787...

      Let
      m = prod(k=1,20,p[k]^a[21-k])
      q = 1516882248248112456504781063232686847
      r = 3014843725938648456075753401380244321
      n = m*q^2
      d = m*(r^2-q^2)/2

      Then we have the primitive equal-tau progression

      tau(n) = tau(n+d) = tau(n+2*d) = a[21] - a[20]

      with tau ~ 8.22215486298630539147326425 * 10^385255
      ln(ln(ln(ln(ln(n))))) ~ 0.9420151824085290419150311

      Proof:

      ? {p=[5,13,17,29,37,41,53,61,73,89,
      97,101,109,113,137,149,157,173,181,193];
      q=1516882248248112456504781063232686847;
      r=3014843725938648456075753401380244321;
      if(2*prod(k=1,20,p[k])^2==q^2+r^2,print(ok))}
      ok

      Primality testing 1516882248248112456504781063232686847
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 100.00%
      and helper 100.00% (400.00% proof)
      1516882248248112456504781063232686847 is prime!

      Primality testing 3014843725938648456075753401380244321
      [N-1/N+1, Brillhart-Lehmer-Selfridge]
      Calling N+1 BLS with factored part 100.00%
      and helper 100.00% (400.00% proof)
      3014843725938648456075753401380244321 is prime!

      I reckon this is big enough, Jon?

      David



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