Puzzle 307. Record Smith Numbers

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Puzzle 307. Record Smith Numbers

About Smith numbers in this site please go to puzzles 107, 108 & 247.

Let's remember five ways of obtaining Smith numbers, and the corresponding known records:

Method * Record

9R_n*Q^S*10^M where R_n is a prime repunit and Q is a palindromic prime of the form 10^2K +A*10^K + 1; by Samuel Yates 9R₁₀₃₁ *(10⁶⁹⁸⁸²+3*10³⁴⁹⁴¹+1)¹⁴⁷⁶*10^3913210 digits = 107,060,074; palprime by Heuer in 2002; Smith by P. Costello

(Heuer also found, later the same year 2002, this palprime, 10^75630+8*10^37815+1. Can you construct the corresponding Smith?)

P*Q*10^M where P was a small prime and Q was a known Mersenne prime, by P. Costello 191*(2²¹⁶⁰⁹¹-1)*10²⁶⁶ , digits =65319 digits, Q by Slowinski 1985; Smith by P. Costello
(Perhaps it's time to get the minimal P for the recently discovered Mersenne 42?)

t*9R_n*10^M where t comes from the set
{2, 3, 4, 5, 7, 8, 15}, by W. McDaniel ?

11^K*9R_n*10^M where R_n is any (prime?) repunit, by Kathy Lewis ?

2*P where P is prime whose digital root is 2, by ?
Application 1: P=A*10^X+1, digital root of A=1, by CR

Application 2: P=B*10^X-1, digital root of B=3, by CR

P=105994 · 10¹⁰⁵⁹⁹⁴+1, digts = 106000, by g157 , 2000, Generalized Cullen; Smith by CR
P=93 · 10⁵²³⁰³-1, digits = 52305, by g243, 2004, Near-repdigit; Smith by CR

* For the methods see the Pat Costello's page

Question:

Can you produce/improve any of these records in its own model?

Larry Soule has found (May 17, 2005) a new record Smith number (7817325 digits!) using the P*Q*10^M scheme by P. Costello, if...he has no mistake in the SOD of M42. See below.

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Larry wrote:

I was working on puzzle and believe I came up with the minimal P to generate a smith number via the P*Q*10^M scheme of P. Costello. For Q=M42=2^25964951-1, P=5, and M=1095, we have N=5*Q*10^1095 the sum of digits is 35178055. The sum of the prime factors is 35170385 (for Q) + 5 (for P) + 1095*(2+5)=35178055.

This was implemented with GMP with some workarounds to sum the digits of such large numbers. I believe this N has 7816230+1095 =7817325 digits!

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But... In the problem 74 of these pages we can read that "SOD(M25964951) = 35170384", according to Andrew Rupinski calculations, on March 12, 2005. The difference is minimal... just one unit.

What is the correct number: 35170384 or 35170385?

We need a third calculation.

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Larry wrote (21-24, May, 2005)

Gack! I checked my code and did 2^p rather than 2^p-1. Sorry for the false alarm.

...

The new solution for the minimal P to generate a smith number via the P*Q*10^M scheme of P. Costello is:

Q=M42=2^25964951-1
P=127
M=4634

N=127*Q*10^4634 with the sum of digits being 35175028. This N has 7820865 digits.

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