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According to time complexity, it takes 6 years for this number to be completely factorized.

2343234349823904823904820349802443332395423549502458934234324342344945424523049234254524524

My friend said he could do it by hand. We made a bet. After a day he gave me 5 primes and said he has done it. Multiplying them out, I found they became the number. Is there something I'm missing? Shouldn't this be theoretically impossible?

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    Nah. We have factorization algorithms and computers for a reason. And who's to say he didn't cheat? http://www.wolframalpha.com/input/?i=factor+2343234349823904823904820349802443332395423549502458934234324342344945424523049234254524524 – 2017-02-01
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    Where did you get the "6 years" figure from? – 2017-02-01
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    https://www.khanacademy.org/labs/explorations/time-complexity – 2017-02-01
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    Somebody tell me why it says 6 years. What am I missing? – 2017-02-01
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    The general question of prime factorization might be hard (we don't know for sure) but some specific cases might take very little time. By hand is still impressive, unless "by hand" includes a calculator with arbitrary precision. – 2017-02-01
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    so I happened to have a number that is specifically not hard? – 2017-02-01
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    I think @SimplyBeautifulArt's observation (that you cannot easily guarantee your friend didn't cheat) is on point. There are two factors of $2$, then two four-digit prime factors, and then one remaining factor that is very large. Verifying that that number is prime seems pretty tedious to do by hand, though not impossible. It's about a $275$-bit number. Running probabilistic primality checks on that would take a while by hand. I can't say whether doing it in a day strains credulity, though. – 2017-02-01
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    But why didn't the calculator have trouble? I thought a computer should take 6 years? – 2017-02-01
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    @Canety: To the extent that that time complexity calculator is accurate, it can only be so up to a constant factor, for the simple reason that this problem will submit to additional processing power: Throw twice the compute power at it, and you will halve the completion time. So unless we know what computing power is assumed by the page you linked to, it is not advisable to take the resulting time too literally. It is the *dependence*, the variation of that time with the number of bits in the number, that is the main message of that page. – 2017-02-01
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    @Canety Simply Beautiful Art's link is only a "partial prime factorization". The factor 5454... is easily verified to be composite (using a fast algorithm such as Miller-Rabin) but factoring it seems to be quite slow – 2017-02-01
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    @SimplyBeautifulArt This is only a partial factorization (and wolfram doesn't claim otherwise). The last factor is composite (as wolfram admits: http://www.wolframalpha.com/input/?i=is+54547552696882185249843318916708824450321892641562440766870835429078592497369018521+prime) – 2017-02-01

1 Answers 1

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Factoring this number by hand is a stretch, but with a modern computer and free software (GP/PARI) it is easy (five minutes on my old laptop):

gp > factor(2343234349823904823904820349802443332395423549502458934234>

%1 = [ 2 2]

[ 1327 1]

[ 8093 1]

[ 5008622079351226528729531 1]

[10890730391051544901210957805049074484498488767350136975291 1]