Re: Two factors of (10^71-1)/9 = R71

• To: mathgroup at smc.vnet.net
• Subject: [mg30132] Re: [mg30125] Two factors of (10^71-1)/9 = R71
• From: BobHanlon at aol.com
• Date: Tue, 31 Jul 2001 04:27:07 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

```In a message dated 2001/7/29 9:42:17 PM, seidovzf at yahoo.com writes:

>The number William Mopppett  wrote:
>           ================
>(10^71 - 1)/9
>is R71, repunit, number with "all ones".
>
>And it has two factors:
>R71 =
>241573142393627673576957439049
>*45994811347886846310221728895223034301839
>
>See, e.g.,
>http://www.ping.be/~ping6758/repunits.htm
>
>But suprisingly enough,
>when,  before looking for "repunit"s in 37.com,
>I asked my PC to work for saturday,
>its Mathematica session was:
>
> \$Version
> 4.0 for Microsoft Windows (December 5, 1999)
>
><< NumberTheory`FactorIntegerECM`
>
>Timing[f = FactorIntegerECM[(2^128 + 1)/f] ]
>{793.95 Second, 59649589127497217} (* good, but...*)
>
>Timing[FactorIntegerECM[(10^71 - 1)/9 ]]
>{112642. Second, \
>111111111111111111111111111111111111111111111111111111111111111\
>11111111} (* ??!! *)
>
>That is Mathematica (or my PC?) considers R71 as prime!
>
>And this is a real suprise
>and hence the question to Mathematica experts:
>how it can be?
>

R71 = 241573142393627673576957439049*
45994811347886846310221728895223034301839;

R71 == (10^71 - 1)/9

True

The documentation for NumberTheory`FactorIntegerECM` states that it extends
Mathematica's integer factoring to all numbers of 40 digits or less.  R71 has
71 digits.  It also states that the algorithm returns a single factor (not
necessarily a prime).  Mathematica recognizes R71 as non-Prime

PrimeQ[R71]//Timing

{0.03333333333284827*Second, False}

Bob Hanlon
Chantilly, VA  USA

```

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