Springer Publishes Crank 'Proofs'

crank, n.
A pejorative term for a person who holds a belief that a vast majority of their contemporaries consider false. A "cranky" belief is so wildly at variance with commonly accepted truth as to be ludicrous, and arguing with cranks is useless, because they will invariably dismiss all evidence or arguments which contradict their unconventional beliefs. (Wikipedia.org)

Two propositions, Fermat's Last Theorem and the Goldbach Conjecture, truly stand out among cranky topics, in drawing the vast majority of bogus "proofs" (although the Riemann Hypothesis may be another contender.) This is well-known among publishers like arxiv.org and philsci-archive, which get plenty of crank submissions. These topics are generally treated with distrust. So it's just stunning that Springer, one of the most trustworthy scientific publishers, may have just printed a book containing crank "proofs" of both these propositions!

The author, Nico F. Benschop of Crank.net fame, has seen basically the same argument refuted by number theorist Robin Chapman seven years ago. (Chapman can be found dismissing this book anew in a sci.math post last week.) You can take a look at the work here through Google Books; on page 133, you'll find the following alleged "proof" of the Goldback conjecture:

Yes, it's three paragraphs of jargon, and it's not clear if it's worth the effort to decipher it. Thanks to John Baez for pointing this out -- Springer really managed to miss a whopper!

Riemann Hypothesis Gets Proved Again

Yet another proof of the Riemann Hypothesis (RH) has been proposed by BYU mathematician Xian-Jin Li. Dr. Li posted his proof on the arXiv a few days ago. (Update, 11:05am.)

Why I'm Not Holding My Breath. Although it's tempting to get excited over a proof of RH, here are a few reasons why I'm not holding my breath:

1. Pitkanen's proof of RH (2001). Withdrawn by author due to errors.


2. Castro & Hahecha's proofs of RH (2001, 2002, 2006). Approach has been rejected.


3. Shi's proof of RH (2003). Proof contains errors.


4. de Brange's proof of RH (2004). Remember all the media attention this got? A counterexample was later produced.


5. Chun-Xuan's proof of RH (2005). Proof contains errors.


6. Aizenberg's counterexample to RH (2007). Withdrawn by author due to errors.


7. Madrecki's proofs of RH (2007a, 2007b, and 2007c). Proofs contain errors.


8. Han's proof of RH (2008). Proof contains errors.

These once-hopeful provers of RH are only the tip of the iceberg. But that's not what's really fearful about proposing a proof of RH. Check out the way that Stanford mathematician Brian Conrad layed into this nut job, for being arrogant about a purported proof!

I suppose you can't escape when duty calls. (A note on Conrad's last sentence: he's evidently not a historian of mathematics.) At any rate, the expert verdict on Dr. Li's proof will be out within a year or so. I'm willing to be patient on this one.

What is the Riemann Hypothesis? The Riemann Hypothesis is that the non-trivial roots of the Zeta function on C, given by


all have real component Re(s) = 1/2. In his famous 1859 paper, Riemann wrote that,

Without doubt it would be desirable to have a rigorous proof of this proposition; however I have left this research aside for the time being after some quick unsuccessful attempts, because it appears to be unnecessary for the immediate goal of my study...

You can read more about RH at Mathworld, about failed proofs at Matthew Watkins's homepage, and about the $1 million prize out for its solution at ClayMath.org.

Edit: Update, 11:05am. Field's medalist Terence Tao has pointed out an error in the proof. It is possible that there are more.