## Monday, August 05, 2013

### Longest time between posing a math problem and it being answered?

(We were asked to remind you: ITCS 2014 Call for papers: call for papers.)

What problem in math had the longest time between POSING IT and SOLVING it? This might not be a well defined question since the notion of when was it posed? might be murky. For some problems even when it was solved? might be murky. Nevertheless I have a candidate:

Is there a straight-edge and compass construction that will, given a square, produce a circle with the same area. (This problem is often called Squaring the circle..)

Wikipedia says that Oenopides was the first person to pose construction problems and that he posed this one. He was born in roughly 500 BC. Even back then there were people who thought it could not be done. However, it was proven impossible when pi was shown to be transcendental in 1882 by Lindemann. (This was one of the motivations for Lindemann.)

This problem was open for roughly 2300 years.

1. Is there any solved problem that was open for longer?
2. Is there any open problem that has been opened for that longer?
3. If you polled people in 400 BC what they would have guessed for which way it would go and when it would be solved?

Will P vs NP take that long?

1. If the problem was posed in 500 BC and proven impossible in 1882 AD, wouldn't it be open for 500 + 1882 = 2382 years, much greater than the 1300 years you claim above?

1. I have fixed it - thanks. I did 1882-500 instead of 1882+500.

2. The problem of the parallel postulate (in modern terms, to prove it from the other axioms of geometry, or to prove it independent of them) must be of similar age. Euclid seems to have been aware of the problem c. 300 BC (he avoids using his 5th axiom if at all possible), and it was solved in 1868 by Beltrami.

3. People have been searching for the best way to multiply integers since the then-1500-year-old document the Rhind Papyrus was a copy of was written. We still don't know, all of written human history later.

4. The twin prime conjecture has been open for approximately 2300 years (the conjecture was made by Euclid around 300 BCE).

1. The twin prime conjecture is not due to Euclid. See

http://mathoverflow.net/questions/7639/twin-prime-conjecture-reference

One could perhaps argue that the question of whether there exist any odd perfect numbers is implicit in Euclid, but he doesn't pose this question explicitly.

5. Garth: it seems like squaring the circle is older by around 200 years
(though we may find that people before Euclid posed Parallel Postulate
and have to revise this) but clearly Parallel Postulate is more important.

Jeff: WOW- complexity of Mult still open!

Yury: Once Twin Primes is solved we'll prob have a new winner of
the open-the-longest-but-solved contest, unless Mult ever gets solved.

6. It's actually Lindemann not Lindermann (without the 'r'). But that doesn't really matter, I guess.

1. Fixed.
Another very old open problem-design a spell checker that is smarter than
its user in terms of proper names.

2. :) i like this idea!

3. :) i like this idea too!

7. How old is the question of whether there is an odd perfect number?

8. the question of the existence of odd perfect numbers is also due to the greeks/euclid. basically number theory & diophantine eqns are the oldest problems. the inherent difficulty of many of them is apparently related to the undecidability phenomenon as in the matijasevich-davis-robbins-putnam proof.
re twin primes note theres been a recent breakthrough by zhang covered in many places.

9. ps in this blog post I ponder the difficulty of P vs NP in a historical context and compare it to scaling olympus mons on mars in contrast/juxtaposition to scaling Everest on earth.

10. The oldest problem is consistency. It was first studied in ancient Mesopotamia, by a Sumerian whose name we do not know. Yet strangely, we know how he looked like. He pondered the question many a day, the entire time sitting on his chair. This fellow was the first royal scribe. He considered himself lucky, he was fast and good with numbers. No one knew of a better scribe. He considered his talent a gift, which allowed him a comfortable life and many luxuries. But he now saw it was also a curse.

Some sunrises ago, his liege and employer asked him to make a count of all the wealth stored in the royal vault, which was comprised of various artifacts and valuables. You see, his liege believed he was the wealthiest man in the world and he wanted to prove it, by measuring his wealth.

For days the scribe and his assistants had counted, making a list of everything in the vault. Then he sat down and multiplied the quantities of everything with their value and then summed it all up, covering a vast amount of tablets with his scribblings. The final amount was extraordinary, never he had written down such a big number. He thought his liege would be much pleased. So he wrote the result all nice and formal in a royal report and presented it to his liege, hoping for a great reward for his endeavors. He did not get was he was expecting. You see, when he presented the report to his lord, the lord sat skeptical and said:

"Very well. But are you sure you have not made a mistake?"

The scribe described how he and his assistants calculated for days. He did every operation twice when multiplying and summing, just to be sure. But his lord did not understand mathematics and so he said to him:

" But are you certain those... calculations of yours are correct? If your methods are false, I shall be ridiculed by all. Begone and do not come back until you can show me that this amount is correct."

The scribe, a noble himself, as his father, knew very well the punishment for disobeying his lord. So he sat in his chair and pondered. He thought for days of how proving his multiplications and additions were correct, but thought as he may, he could not find a solution. He perished in that very chair, by starvation and worry. We may never learn his name, but he was forever immortalized by a sculptor of that court, who was fascinated by the mathematician who stood there, for days, pondering his seemingly inescapable situation.

[The thinking scribe: http://cache2.allpostersimages.com/p/LRG/37/3795/9DIIF00Z/posters/statue-of-a-sumerian-scribe.jpg]

His replacement was his assistant, a good scribe, but he lacked the talent of his master for calculations. However, what he lacked in talent, he replaced with wit. For you see, when the lord asked him for an answer to the same question, the young scribe said:

"We asked the oracle for an omen, my Lord. The Gods confirm everything is in order and this is your true wealth."

[The above story is of course, fictional.]

1. I'm curious- its ficitional in that YOU made it up,
or its a legend that, while false, is... out there and
the reason for the statue?

11. The Biblical prophet Job's (serious) question "Why do the righteous suffer?" has remained open for at least the past four millennia.

Until recently, that is! Now David Deutsch is (seriously) proposing (in The Beginning of Infinity: Explanations That Transform the World, page 212) the simple (too simple?) STEM-centric (too STEM-centric?) answer "All evils are caused by insufficient knowledge".

It will be interesting to see whether the 21st century's exponentiating accumulation of STEM-related knowledge results in an appreciable global reduction of suffering.

Summary  So far, so good! :)

12. Oenopides of Chios was an ancient Greek mathematician and astronomer, who lived around 450 BCE. He was born shortly after 500 BCE on the island of Chios, but mostly worked in Athens.