I have recently been going through my book Maths 1001 making updates for a forthcoming foreign edition (of which more in future). So I have been looking over mathematical developments since approximately 2009. Thus, I present ten major developments in the subject since around then, arranged arbitrarily in ascending order of top-ness.

** 10. Mochizuki’s claimed proof of the abc conjecture.** The countdown kicks off on an awkward note. If Shinichi Mochizuki’s 2012 claimed proof of the abc conjecture had gained widespread acceptance, it would definitely top this list. As it is, it remains in limbo, to the enormous frustration of everyone involved.

**9. The weak Goldbach conjecture.** “From 7 onwards, every odd number is the sum of three primes.” We have known since 1937 that this holds for all large enough odd numbers, but in 2013 Harald Helfgott brought the threshold down to 10^{30}, and separately with David Platt checked odd numbers up to that limit by computer.

**8. Ngô Bảo Châu’s proof of the Fundamental Lemma.** Bending the rules to scrape in (time-wise) is this 2009 proof of a terrifyingly technical but highly important plank of the Langlands Program.

**7. Seventeen Sudoku Clues.** In 2012, McGuire, Tugemann, and Civario proved that the smallest number of clues which uniquely determine a Sudoku puzzle is 17. (Although not every collection of 17 clues yields a unique solution, their theorem establishes that there can never be a valid Sudoku puzzle with only 16 clues.)

**6. The Growth of Univalent Foundations/ Homotopy Type Theory.** This new approach to the foundations of mathematics, led by Vladimir Voevodsky, is attracting huge attention. Apart from its inherent mathematical appeal, it promises to recast swathes of higher mathematics in a language more accessible to computerised proof-assistants.

**5. Untriangulatable spaces.** In sixth position is the stunning discovery, by Ciprian Manolescu, of untriangilatable manifolds in all dimensions from 5 upwards.

**4. The Socolar–Taylor tile.** Penrose tiles, famously, are sets of tiles which can tile the plane, but only aperiodically. It was an open question, for many years, whether it is possible to achieve the same effect with just one tile. Then Joan Taylor and Joshua Socolar found one (pictured above).

**3. Completion of the Flyspeck project.** In 1998, Thomas Hales announced a proof of the classic Kepler conjecture on the most efficient way to stack cannon-balls. Unfortunately, his proof was so long and computationally involved that the referees assigned to verify it couldn’t complete the task. So Hales and his team set about it themselves, using the Isabelle and HOL Light computational proof assistants. The result is not only a milestone in discrete geometry, but also in automated reasoning.

**2. Partition numbers.** In how many ways can a positive integer be written as a sum of smaller integers? In 2011, Ken Ono and Jan Bruinier provided the long-sought answer.

**1. Bounded gaps between primes.** It’s no real surprise to find that the top spot is taken by Yitang Zhang’s wonderful 2013 result that there is some number *n*, below 70 million, such that there are infinitely many pairs of consecutive primes exactly *n* apart. The subsequent flurry of activity saw James Maynard, and a Polymath Project organised by Terence Tao, bring the bound down to 246.

## But, but,…?!

Where’s Hairer’s work on the KPZ equation? What about Friedman’s new examples of concrete incompleteness?! What can I say? It’s just for fun, folks. If you think I’ve got it horribly wrong, then feel free to compile your own lists. (The real answer for such things being left out is that I couldn’t easily update my book to include them.) And now…

## Bonus feature! Progress in computational verifications and searches

In no particular order:

- The simple continued fraction of π has now been computed to the first 15 billion terms by Eric Weisstein, up from 100 million.

- The decimal expansion of π has been computed to 13.3 trillion digits, up from 2.69999999 trillion.

- The search for the perfect cuboid: if one exists, one of its sides must be at least 3 trillion units long, up from 9 billion.

- Goldbach’s conjecture has been verified for even numbers up to 4 × 10
^{18}by Oliveira e Silva, up from 10^{18}.

- The largest known twin primes are the pair either side of 3756801695685 × 2
^{666669}, up from 2003663613 × 2^{195000}.

- The largest known prime, and the 48th known Mersenne prime (up from 47), is 2
^{57885161}-1, up from 2^{43112609}-1.

- The Encyclopedia of Triangle Centres contains 7719 entries (up from 3587).

- The longest known Optimal Golomb Ruler is now 27 notches long (up from 26):

(0, 3, 15, 41, 66, 95, 97, 106, 142, 152, 220, 221, 225, 242, 295, 330, 338, 354, 382, 388, 402, 415, 486, 504, 523, 546, 553)

- The most impressive feat of integer-factorisation using classical computers, is that of the 232-digit number RSA-768:
1230186684530117755130494958384962720772853569595334792197322452 1517264005072636575187452021997864693899564749427740638459251925 5732630345373154826850791702612214291346167042921431160222124047 9274737794080665351419597459856902143413

into two 116 digit primes

3347807169895689878604416984821269081770479498371376856891243138 8982883793878002287614711652531743087737814467999489

and

3674604366679959042824463379962795263227915816434308764267603228 3815739666511279233373417143396810270092798736308917.(The previous record was the 200 digit semiprime RSA-200.)

- The most impressive feat of integer-factorisation using a quantum computer is that of 56,153=233 × 241. The previous record was 15.

- The Collatz Conjecture has been verified for numbers beyond 2 × 10
^{21}. The previous record was 5.76 × 10^{18}. (However, this has happened via a patchwork of distributed computing projects, and I have not been able to establish with any certainty that*every*number up to the new higher limit has been checked. I encourage someone in this community to organise all the results in a single location.)

The formal proof of the odd-order theorem? Fixing mistakes I believe had still been in there, despite decades of work simplifying it.

I hadn’t realised it needed fixing as well as formalising. How long till the full classification of finite simple groups goes the same way…? Of course, it’s a tremendous achievement which fully deserves inclusion.

Well, they were subtle ones that people didn’t know about 🙂 As for the CFSG, I don’t know. As a guess, I would say a couple of decades, barring amazing developments in proof assistants, and assuming continual and mildly increasing work on it. To be honest, I don’t even know what is being done on it right now!

Since you mentioned integer factorization records, I would add the amazing progress (both theoretical and computational) on the discrete logarithm problem in finite fields:

https://en.wikipedia.org/wiki/Discrete_logarithm_records#Finite_fields

https://ellipticnews.wordpress.com/2013/06/21/quasi-polynomial-time-algorithm-for-discrete-logarithm-in-finite-fields-of-smallmedium-characteristic/

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Reblogged this on Comentários, Críticas, Dicas etc. and commented:

Just another top ten list.

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As far as I know, the Collatz Conjecture verification entry should not be on the list of top computational verifications. The Sontag’s website is very vague about what as been achieved.

Certainly many integers have been tested, but it appears that they start at a certain initial value

(close to 2,361,184,410,093,970,784,932), which raises the obvious question: “have all lower

values been tested?” Unfortunately, at least to me, that seems unlikely.

Tomás, I broadly agree with your comment and said something about this in the post. But I hope the situation is not so bad as you say. I emailed Jon Sonntag about this point, and he explained that there was an earlier project “named 3x+1@home which ended in 2008 and was run by Markus Tevooren”. I think it is possible, maybe even likely, that between them the projects have covered all lower values. But of course we would like to be able to say so with more confidence! Thank you for your comment.

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Reading your 1001 mathematics, I noticed the absence of the Parker-Sochacki solution to the Picard iteration. Because of its importance to population dynamics analysis and the N-body problem of celestial mechanics, I thought I should bring it to your attention.

I must admit that this wasn’t discovered in the last five years; its first discovery was probably more than a century ago, and most recently discovered by someone working on the atomic weaponry of the US, an italian, and (most recently) Ed Parker and James Sochacki, published in Neural, Parallel, and Scientific Computations 4, 1996.

What does this do? It is incredibly effective at generating maclaurin Series solutions to systems of IVP equations and differential equations.

You can see more at wikipedia, or you can find a little tutorial at

http://arxiv.org/pdf/1007.1677?origin=publication_detail