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Internet, Dogs, and the ABC Conjecture

May 14, 2019

An inappropriate comment on the ABC conjecture

Joseph Oesterlé and David Masser are famous for their independent discovery of the ABC conjecture.

Today I want to point out an unfair comment about their discovery.

Anonymity on the Internet was captured by a famous 1993 cartoon in the New Yorker magazine titled, “On the Internet, nobody knows you’re a dog.” Amazing to think that was more than a quarter-century ago and remains true. But people can tell if what you’ve written is something inappropriate.

The Comment

The comment is:

SAYS WHO??? I have some trouble with this item.

Masser is a Fellow of the Royal Society, who was elected in 2005. He is

{\dots} also responsible, following an earlier insight of Joseph Oesterlé, for formulating the abc conjecture; this is a simple statement about integers which seems to hold the key to much of the future direction of number theory.

See this link for his full citation and the comment. Click on the show more bibliography button there. The comment is apparently anonymous, although the author is probably known to some. I thank Joël Ouaknine for pointing out this strange comment.

Update: Ken speculates that it’s a misplaced comment by an editor of the Royal Society website itself. Perhaps they compose HTML from MS Word or Acrobat or other software that provides comment bubbles—but this one escaped the bubble and wasn’t noticed. Editors of Wikipedia have automatic tools for flagging assertions that are unsupported or at least need citation.

What the comment undoubtedly shows is vigorous debate behind the walls of Britain’s august institution. So let’s say a little more on what the comment is about.

The ABC Conjecture

The biggest mysteries about numbers often concern the interaction between addition and multiplication. For example:

  • The twin prime conjecture: There are an infinite number of primes {p} such that {p+2} is also prime. This is due to Alphonse de Polignac.

  • The Goldbach conjecture: Every even number greater than four is the sum of two odd primes. This is due to Christian Goldbach.

  • The Brocard conjecture: There are only a finite number of solutions to {n! = m^{2} + 1} in natural numbers. This is due to Henri Brocard.

Suppose that {A + B = C} where {A,B,C} are positive and co-prime natural numbers. Let {D} be the product of all the distinct prime divisors of {ABC}. Then the ABC conjecture says that

\displaystyle  C \le O(D^{2}).

Note, this inequality does indeed connect adding with multiplying. The usual conjecture is stronger, see this for details.

The ABC conjecture appears to be open, even though Shinichi Mochizuki has claimed a proof for years. See this for a discussion about the status of the conjecture.

Despite multiple conferences dedicated to explicating Mochizuki’s proof, number theorists have struggled to come to grips with its underlying ideas. His series of papers, which total more than 500 pages, are written in an impenetrable style, and refer back to a further 500 pages or so of previous work by Mochizuki, creating what one mathematician, Brian Conrad of Stanford University, has called “a sense of infinite regress.”

The Comment II

The comment on Masser’s work is wrong, strange, inappropriate. Oesterlé and Masser deserve more credit, not less, for their brilliant discovery of the ABC conjecture. There are now many—perhaps hundreds—of applications of the ABC conjecture. For example consider generalizations of Fermat’s Last Theorem. Suppose that

\displaystyle  x^{p} + y^{q} = z^{r} \text{  (*)}

where {p,q,r} are odd primes. And r > 6. Provided {x,y,z} are positive and co-prime, it follows by the ABC conjecture that {z^{r}} is bounded by {O(z^{2})}. This is impossible for {z} large enough since {r \ge 3}. Therefore, (*) can only have a finite number of solutions. Pretty neat.

Open Problems

Do you know of any other inappropriate comments of this kind?

[added remark by Ken, linked rather than embed dog cartoon]
[Added prime r must be 7 or larger. Thanks to comment by MadHatter.]

17 Comments leave one →
  1. Serge permalink
    May 14, 2019 11:01 am

    Surely Masser’s bio will get cleaned before anybody can tell if Mochizuki’s proof is correct. Mathematics isn’t always an exact science…

  2. rjlipton permalink*
    May 14, 2019 11:09 am

    Dear Serge:

    Pretty funny comment. In the spirit of todays piece. Mathematics isn’t always an exact science…I like this quote. It really is not exact at all.


    • Serge permalink
      May 14, 2019 2:38 pm

      Thank you! On Wikipedia they use the tag “citation needed” instead of “says who”, but obviously the ABC conjecture doesn’t need it.

  3. May 14, 2019 11:14 am

    On the internet, people eventually figure out you’re a dog.

  4. MadHatter permalink
    May 14, 2019 6:49 pm

    I don’t understand the application to FLT. Why is D, the product of all the distinct prime divisors of x^p \cdot y^q \cdot z^r, bounded by O(z)?

    • rjlipton permalink*
      May 14, 2019 8:06 pm

      Dear MadHatter:

      Great name. I made an error in my calculation. Oops. The value of D is bounded by xyz. So z^r is at most order O((xyz)^2) by the ABC conjecture. This is bounded by O(z^6). So we need that r is at least 7. I will fix this soon.


  5. May 15, 2019 4:45 am

    I contacted the Royal Society and they “have fixed the biography and are investigating the issue as a matter of urgency”…

    • rjlipton permalink*
      May 15, 2019 7:25 am

      Dear defjaf:

      Thanks for this. The comment is gone.


  6. pzbornik permalink
    May 15, 2019 1:36 pm

    Well this comment seems to have been there at least since 2015 – here is the original version with “SAYS WHO???” –

    • rjlipton permalink*
      May 15, 2019 3:43 pm

      Dear pzbornik:

      Thanks for finding the original link. I thought they knew about the issue. Well it is fixed now.


  7. Peter Gerdes permalink
    May 15, 2019 11:30 pm

    Perhaps in the spirit of Ken’s suggestion I’d imagine that this might even be Masser himself. If this is anything like a faculty bio in the states it wouldn’t be unusual to ask the subject themselves to supply something to post. Masser may have simply accidently submitted a draft that included a note to himself about adding a potential citation.

    In any case I’d be shocked if this was a comment on Masser’s work rather than a simple note about adding a citation that accidently got left in.

    • rjlipton permalink*
      May 16, 2019 8:41 am

      Dear Peter Gerdes:

      I read it as a not so nice comment. But who knows. I thought about discussing in more detail how Masser does not get enough credit for this great conjecture. Perhaps another time.

      Thanks again for comment. You could be right.


  8. David Spector permalink
    May 17, 2019 11:07 am

    Web fora are filled with random comments by untrained people who offer their incorrect opinions on an apparently infinite variety of subjects. Some of these people do it to elicit replies, Whether the replies are favorable or unfavorable, they apparently bolster the poster’s ego. There is no point in rising to the bait and wasting time with such comments. In this case, the comment appears to be just this sort of thing.

    • rjlipton permalink*
      May 17, 2019 11:13 am

      Dear David Spector:

      What exactly are you referring to as the “comment”? Is it the comment on the Royal site? Any way I do not mind answering comments.


      • David Spector permalink
        May 17, 2019 11:25 am

        I feel very humble to have received a reply from the great R J Lipton. I just meant comments like “SAYS WHO??? I have some trouble with this item.” We may not mind answering comments like this, but we should not encourage such brief, unsupported, and wild opinions, in my opinion… I came to your blog because of Richard M. Karp’s wonderful Simons Foundation lecture. By the way, I love imagining great, simple approximate solutions to the TSP just for the fun of it. But I would never be so arrogant as to publish such a “solution”.

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