We know that the number of distinct roots, N0, of a polynomial cannot be greater than its degree, but of course it could be much less - many of the roots could be repeated. Perhaps surprisingly however, it seems there is something about adding two polynomials with many repeated roots together which prevents their sum from having the same property:
Mason-Stothers Theorem. Let f,g,h be relatively prime polynomials with f+g =h. Then N0f g h > maxdegf, degg, degh.
For example, when there is a triple of relatively prime polynomials of degree n which sum to 0, then they must have at least n+1 roots between them. Although this theorem admits an elementary proof, it was only stated and proved as late as 1981.
The abc Conjecture
The abc conjecture is essentially an analogue of the Mason-Stothers Theorem for integers. Roughly, the idea is that if two integers are products of large powers of small primes, then their sum should be a product of small powers of large primes. To translate the theorem more precisely, we replace the number of distinct roots of a polynomial by the radical of an integer, that is, the product of its prime factors, radn = ∏p|n p ; and we replace the degree of a polynomial by the absolute value of an integer. So the precise analogue would be:
For any triple of coprime positive integers a,b,c with a+b =c, rada b c > c. (This is not actually true!)
In fact there are known to be infinitely many examples of such triples with rada b c < c. However, the actual conjecture states that the previous statement is almost true, in the following sense. If we just raise the threshold to the power 1+ε, no matter how small we make ε, then there are only finitely many counterexamples:
abc Conjecture. For every ε > 0, there are only finitely many triples of coprime positive integers a,b,c with a+b =c and rada b c1+ε < c.
The abc conjecture was proposed by Oesterlé and Masser in 1985, and it has since been determined to be both extremely difficult to prove, and at the same time very important! Indeed, it has been shown that if the abc Conjecture is true, then a number of important results in number theory, including Fermat's Last Theorem, would follow as consequences.
In 2012, Mochizuki published a claimed proof of the abc conjecture. Although there have been many claims to the proof previously, this one is deemed to be more serious as it comes from a renowned mathematician. However the proof introduces so much new mathematics that it remains as yet unverified by the mathematics community due to its complexity.
An alternative formulation
For a given value of ε, there would have to be a maximum value, say Cε, for ratio crada b c1+ε for the conjectured finitely many triples a,b,c satisfying rada b c1+ε < c. This allows us to rewrite the abc Conjecture as follows:
abc Conjecture. For any real number, ε>0, there exists a finite constant Cε such that for any three relatively prime integers a, b, c which satisfy
a+b = c,
c < Cε rada b c1+ε .
The quality of a triple a,b,c
The quality of a triple of coprime positive integers a,b,c with a+b=c is defined to be the ratio
qa, b, c = lnclnrada b c.
In terms of this new concept we can rewrite the abc Conjecture as follows:
abc Conjecture. For every ε > 0, there are only finitely many triples of coprime positive integers a,b,c with a+b =c and qa, b, c >1 + ε.
In other words, triples of high quality are rare, and the higher the quality, the rarer they get. Here is the list of integer triples with a, b ≤ 200 and quality greater than 1:
qa, b, c
Note that the values of a and c are all prime powers, and each b has at most two prime factors. Note also that due to the easy to prove formula
rada⋅b⋅c ≤rada⋅b⋅a+b ≤a⋅b⋅a+b ≤a+b34=c34,
it follows that the quality of an integer triple a,b,c is bounded below by 1/3.
Relation to Fermat's Last Theorem and the Pythagorean Theorem
As seen above, integer triples of high quality tend to be comprised of integers with relatively few prime factors, each raised to a relatively high power. This recalls the Diophantine equation
a n+b n=c n.
The Pythagorean Theorem deals with the case n=2:
Pythagorean Theorem. A triple a,b,c satisfies a2+b2=c2 if and only if a,b,c are the lengths of the sides of a right angle triangle with the right angle between sides of length a and b.
There are in fact infinitely many such Pythagorean triples, and they are easy to parameterize.
Fermat's Last Theorem states that when n > 2 this equation admits no solutions in positive integers:
Fermat's Last Theorem. If n > 2, the equation a n+b n=c n admits no positive integer solutions a,b,c.
Although Fermat himself claimed a proof in 1637, he did not present it, so the theorem remained officially unproven until a proof by Wiles in 1994, some 357 years later!
Roughly speaking, the abc Conjecture suggests that the equation a+b=c has few solutions with a,b,c all being products of few prime factors, raised to high powers. Fermat's Last Theorem says that when these powers are all the same, there are no such solutions.
Adjust the sliders or type in the values directly to choose values for mina,b, maxa,b, and a. mina,b and maxa,b determine the horizontal bounds of the viewing window, as well as restrict the possible values of a and b. Click on the plot to choose a value of b; information about the quality of the triple a,b,c will be shown. What is the highest quality triple you can find?
Note: If you set the value of mina,b and maxa,b too high, computing the qualities of the triples (a,b,c) could take a long time.
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