I would like to thank Chandrakant Sakharwade for introducing me to this subject.
This VERITAS is about a very deep and interesting relationship regarding prime numbers. In 2001 I had written a VERITAS article on prime numbers: http://unvarnished-veritas.posterous.com/the-prime-adventure
Before I start Let me say that to read this VERITAS you may have to concentrate and maybe reread certain sections. I had to read several articles several times before I could understand the abc conjecture. I will try to make it as simple for you as possible by giving many examples and proceeding in a step by step fashion.
Before I talk about the conjecture, let me give you an overview of its importance. It is generally considered one of the most important unsolved problems in mathematics. There are a lot of other problems in number theory which would be solved almost effortlessly if the abc conjecture was proven. One of them is Fermat’s last theorem. We all know that Fermat’s last theorem was proved by Andrew Wiles in 1995. That was a very complex proof- over a 100 pages long. Some people have suggested that if abc conjecture is proven then Fermat’s last theorem can be proven in less than 1 page! And there are many important unsolved problems which would be solved as soon as we have a proof of the abc conjecture.
Now, let me explain what the conjecture is all about.
Lets start with the simple equation: a + b = c. a,b and c are positive integers and should have no common factors except 1. So there should be no number except 1 which can divide the three numbers. To give you an example: 2 + 3 = 5 can be such an equation. 5,3 and 2 do not have any common factor. Another equation can be 1+ 2 = 3. 1,2 and 3 do not have any common factors. Such numbers which do not have any common factor are called co-prime numbers.
Now lets take the numbers a, b and c and multiply them to form a number d. Now a very important theorem of mathematics( called The Fundamental theorem of Arithmetic) states that any number is either a prime or can be expressed as a product of primes. Take the number 17 – it is a prime. Take 18- it can be expressed as 2X 3X 3 ( and 2 and 3 are primes).
So we take the number d ( which is the product of a, b and c) and find its prime factors- i.e express it as a product of primes. So for our equation 1+2 = 3 d is 1X2X3 = 6 and 6 can be expressed as 2 X 3. Similarly for our equation 2 + 3 = 5, d is 5*3*2 = 30 and 30 can be expressed as 2 * 5 * 3. Lets take another equation 4 + 5 = 9 ( you can check that 4, 5 and 9 do not have any common factors). For this equation d is 4 * 5 * 9 = 180. And 180 can be expressed as 2 * 2 * 3 * 3 * 5 and we know that 2,3 and 5 are all primes.
Now we find the radical of d. What this means that in the prime factors of d remove any prime that occurs more than once and multiply the rest.
So in the equation 1+2 = 3, d is 6 and 6 is 2 X 3 – there is no duplicate primes so the radical of 6 is 6. We write rad(6) = 6. For the equation 2 + 3 = 5, d is 30 and 30 can be expressed as 2 * 3 * 5 – again there are no duplicates so rad(30) = 30. Now lets take our equation 4 + 5 = 9. For this d is 4 * 5 * 9 = 180. And 180 can be expressed as 2 * 2 * 3 * 3 * 5. Here we have duplicates. Remove them – there are two 2s and two 3s. Remove the duplicates in the product so rad(180) is 2 * 3 * 5 = 30.
So we have three examples:
1+2 = 3 . d is 6 and rad(d) is 6
2+3 = 5 . d is 30 and rad(d) is 30
4 + 5 = 9. d is 180 and rad(d) is 30
In all the above cases rad(d) is greater than c. For example in the last equation rad(d) is 30 and c is 9. So rad(d) > c.
Can it ever happen that rad(d) is less than c? Yes it can be. Lets take an example:
5 + 27 = 32
Now d is 5 * 27 * 32 = 4320. The unique primes in 4320 are 2,3 and 5 which makes rad(4320) = 30 which is less than c which is 32.
So there are cases in which rad(aXbXc) is less than c. However these cases are rare. The funny thing is that they are infinite yet rare. For c < 10000 there are only 120 a,b,c triplets in which rad(d) or rad( aXbXc) is less than c.
The abc theorem is about these rare, yet infinite abc triplets.
So c can be greater than rad(d). Can c be greater than d squared? Maybe – though we have not found any abc triplet that has this property. There are many examples for abc triplets in which c is greater than d to the power of something between 1 and 2. For example for our example with 5 + 27 = 32 , c is 32 and rad d is 30. So c is greater than rad(d) to the power 1.018
Take another example: 1+ 8 = 9. d is 72 . 72 can be written as 2 * 2 * 2 * 3 * 3 . So rad (72) is 6. And 6 is less than c which is 9. So we have a abc triplet in which rad(d) is less than c. So c is greater than rad(d) . c is even greater than d ^ 1.2262 in this case.
So once again: There are infinitely many abc triplets. Yet abc triplets are rare!
Now after this background( phew!) we come to the abc conjecture:
For any number t greater than 1 there are a finite number of abc triplets which satisfy : c > [rad(d)] ^ t
So in plain English, for any number t which is greater than 1, you can find limited or finite number of abc triplets so that c is greater than rad( aXbXc) raised to power t. So you can take any number t greater than 1: say you take 1.2234. There would be a finite number of abc triplets in which c is greater than rad(abc) ^ 1.2234. For t >2 there may be no such triplets.
Note the most interesting part: I said that abc triplets are infinite yet rare. So there are an infinite number of triplets which satisfy c > [ rad(d)] ^ 1 but only a finite number of triplets which satisfy c > [ rad(d)] ^ 1.01 or c > [ rad(d)] ^ 1.001 or c > [ rad(d)] ^ 1.0000001 or rad(d) raised to the power of 1 + the tiniest possible thing that you can imagine. So the infiniteness suddenly becomes finite even for a tiny variation from 1 to 1.01 or 1.00001 or 1.0000000001! That is the AMAZING part! So before you my dear friends lies the wonderful boundary between the finite and the infinite!
Well it might seem that mathematicians are getting crazy about details. But this is indeed a very important theorem. This conjecture was proposed by David Masser and Joseph Oesterle in 1985. And since that day there has been a huge excitement and work by mathematicians around the world to prove this. And the exciting news is that last week Shinichi Mochizuki of Kyoto University has claimed to have solved it. It is a 500 page paper – incredibly interesting and suspenseful, I am sure. 🙂
However note that in science we should not go by the first reports. There would be verifications and discussions before this paper is accepted. In science there is always a process of testing by different teams before a fact is considered to be reasonably accepted as true. However the press seems to jump to conclusions when the first paper on a subject is presented. And that is why people think that scientists keep claiming different things every month. On some days you will see a newspaper report that coffee is good for you and then a few days later, it is the worst thing in the world. The problem is not science or its method – the problem is the newspaper reporters who want “exciting” science news to fill their paper and so will publish something even before it is verified through hundreds of experiments by independent teams.
Finally before I end, here is an interesting poem on prime numbers. Before you read the poem let me tell you what the stanzas refer to. The first stanza is about the fact that nothing can divide them but they can form all other numbers( see fundamental theorem of arithmetric described above). The second stanza tells us that in the sequence of natural numbers, prime numbers appear unexpectedly. The third paragraph refers to the fact that in the sequence of natural numbers, first there are many primes but as the numbers get larger, the primes get more and more rare and less dense. The last stanza refers to the fact that mathematicians have tried hard to find a formula for prime numbers but have all failed. You can find all these facts in my 2001 VERITAS article on prime numbers: link at the top of this article.
Let Us Now Praise Prime Numbers
by Helen Spalding
Let us now praise prime numbers
With our fathers who begat us:
The power, the peculiar glory of prime numbers
Is that nothing begat them,
No ancestors, no factors,
Adams among the multiplied generations.
None can foretell their coming.
Among the ordinal numbers
They do not reserve their seats, arrive unexpected.
Along the lines of cardinals
They rise like surprising pontiffs,
Each absolute, inscrutable, self-elected.
In the beginning where chaos
Ends and zero resolves,
They crowd the foreground prodigal as forest,
But middle distance thins them,
Far distance to infinity
Yields them rare as unreturning comets.
O prime improbable numbers,
Long may formula-hunters
Steam in abstraction, waste to skeleton patience:
Stay non-conformist, nuisance,
To system, sequence, pattern or explanation.
Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun