“The essence of mathematics lies in its freedom.” - Georg Cantor
When it comes to solving problems, mathematics likes to use theorems and formulae. Once proven, they’re regular used to serve certain types of problems. Before theorems become theorems, they’re referred to as conjecture.
A conjecture in mathematics needs to prove a certain mathematical property but can still be used when no solid disproof has been shown. Mathematicians can still discover and create new rules. Essentially, conjecture is much like a hypothesis, but a hypothesis is something that can be tested to find the answer, which isn't always the case with conjecture.
Despite how clever mathematicians are, there are still quite a few conjectures out there. Even centuries of research and advances in technology have amounted to nothing.
Ready to become a real mathematician?
In this article, Superprof is looking at conjectures and a brief history of mathematics.
What Is a Conjecture?
According to Wikipedia, a conjecture is “a conclusion or a proposition which is suspected to be true due to preliminary supporting evidence, but for which no proof or disproof has yet been found.”
Conjectures, though unproven, are sometimes used as if they are.
There are three ways to resolve conjecture:
- Independent conjectures
This is true of all fields of mathematics from geometry to cryptography and every problem presented will need to be resolved in one of these three ways before it can stop being conjecture.
If you’re at school and are looking for help with your maths, consider getting in touch with a private tutor. As you’ll see, logic is an important part of maths. Making logical deductions is also an important part of your everyday life.
Now let’s see how to prove a conjecture.
How to Show if Mathematical Conjecture is True
Are you familiar with algebra?
This is a form of mathematics when you use letters instead of numbers to represent unknown values. Most of us will first start seeing algebraic problems once we're secondary school students and will regularly be given equations to solve.
With a problem or conjecture, algebra is often an essential part of mathematical proof as you need a fictitious or theoretical version of your proof without specific numbers. This is how you can prove that it works in any situation.
Algebra is a great way to prove a conjecture and to prove it, you need three examples. To disprove it, you only need one example.
Let's start with some very simple equations with a known answer and test them as conjecture.
“Pick a number and subtract 3. Double it and add 6.”
Let’s say that N=5.
- 5 - 3 = 2
- 2 x 2 = 4
- 4 + 6 = 10
Let’s say that N=7.
- 7 - 3 = 4
- 2 x 4 = 8
- 8 + 6 = 14
For the last example, let’s say that N=30.
- 30 - 3 = 27
- 2 x 27 = 54
- 54 + 6 = 60
In each example, the final result is double the original. This is our conjecture and to solve the problem, we need to prove that it's always the case and not just with the numbers we picked.
Since algebra can replace a number with a variable, we can show that it’s true:
- 2(N-3) + 6
- = 2N - 6 + 6
- = 2N
We can establish a mathematical rule that states “When you pick a number, subtract three, double the result, then add 6, you’ll get double the starting number”.
This is because the problem not only works with specific numbers, but it can also work with any unknown number thanks to our algebraic proof.
All great mathematicians have had to prove their theorems and it isn’t always easy. This can take anywhere from hours to years of work!
Here are some of the most famous mathematical theorems:
- Gauss’ Theorem
- Pythagoras’ Theorem
- Fermat’s Last Theorem
- Thales’ theorem
- Gödel's Incompleteness Theorems
Basically, conjecture becomes a theory once it can be proven.
As obvious as some may seem, the proof may be harder to find.
Let’s have a look at some famous examples.
Famous Mathematical Conjectures
Let’s start with Euclid’s perfect numbers. These are positive integers that are equal to the sum of its positive divisors. Euclid found four of them: 6, 28, 496, and 8128.
We currently are aware of 51 examples. But there are two unanswered questions.
“Is there an infinite number of even perfect numbers?”
“Are there odd perfect numbers?”
Nobody knows, not even the world’s greatest mathematicians. This is one of the problems that remain unsolved and even though people have spent a lot of time working on creating equations and formulae to prove it, no mathematician has managed it yet.
Here are some other famous conjectures.
This conjecture dates back to 1742. It’s one of the best-known problems in number theory and still hasn’t been solved.
It shares similarities with the Riemann hypothesis and the twin prime conjecture.
Goldbach’s conjecture states:
“Every even whole number greater than 2 is the sum of two prime numbers”.
So 2N = p + q.
2N is always an even number and p and q are two primes.
As a reminder, a prime number is only divisible by 1 and itself. The first primes are: 2, 3, 5, 7, 11, 13, 17, 19, etc.
The conjecture is still unproven. It’s been proven up to 4 x 1018 for even numbers.
Fermat’s Last Theorem
Do you remember Pythagorean triples?
Fermat's Last Theorem, also known as Fermat's conjecture, is based on these famed Pythagorean triples.
By accepting that a2 + b2 = c2 (which you’ll see for right triangles), would it still work if the powers were changed to something other than zero?
It deduced that “It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers.”
Fermat’s Last Theorem can be expressed as:
“For any integer n > 2, the equation an + bn = cn has no positive integer solutions”
Pierre de Fermat originally said that he had proof for the problem, but it wouldn't fit in the margin. Many aspects of other problems that Fermat had suggested were later proven, but the last theorem remained an unanswered question in mathematics for three and a half centuries.
Eventually, Andrew Wiles was able to provide proof for the problem and would open up new approaches in related branches of mathematics by doing so.
This conjecture states:
“For all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.
This followed the logic of Fermat’s Last Theorem.
It was disproven in 1966 by Lander and Parkin.
This conjecture deals with geometric topology. It was proven by Perelman in 2003.
“Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.”
That said, this is a level of maths that’s probably too complex to get into for the amateur mathematician.
The Riemann Hypothesis
When it comes to problems in maths that remain unsolved, The Riemann Hypothesis is probably the most famous. The conjecture states that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2.
The reason this hypothesis is so important is that mathematicians, especially those working in number theory, will be able to learn a lot about the distribution of primes.
However, despite being proposed by Bernhard Riemann way back in 1859, it remains unsolved over a century and a half later. Hopefully, one day, an incredible mathematician will find the answer to this important problem.
Discover different Maths tuition on Superprof.
Hopefully, you should now know a little bit more about what conjecture is in mathematics.
You can try it out with your own basic ideas. Use logic to prove your conjectures.
If you'd like to learn more about maths, consider getting help from one of the many talented and experienced tutors on the Superprof website.
You can find tutors specialising in maths for all levels from primary school to university. There are different ways to learn from a private tutor so make sure you choose the type of tutoring that works for you, how you like to learn, and your budget.
Don't forget that many of the tutors on Superprof offer the first hour of tutoring for free so you can try a few out before deciding on which one is right for you. You could also try out the different types of tutoring if you're not sure which one you'd prefer.
It's always a good idea to outline your requirements before you start looking for tutors. On the Superprof website, you can see what experience they have, what their other students have to say about them, and how much they charge each hour. Before you start getting in touch with tutors and arranging free lessons, we recommend that you narrow down your search to tutors that meet your requirements.