Articles
BY OUR STUDENT CONTRIBUTORS
Sayonee Das As far back as the time of the ancient Greeks, mathematicians have studied whole numbers and their properties. This branch of maths is called number theory, and is one of the oldest mathematical fields. To this day, there are open questions in number theory, which demonstrates its complexity.
One source of such open questions is the study of perfect numbers. A perfect number is a whole number (integer) which equals the sum of its proper divisors. For example, 6 is divisible by 1, 2, and 3 and it’s also equal to their sum as 1 + 2 + 3 = 6. Likewise, 28 is divisible by 1, 2, 4, 7, and 14 and it’s equal to 1 + 2 + 4 + 7 + 14 = 28. The simplicity of what defines perfect numbers makes them easily understandable to non-mathematicians, so their study intrigues mathematicians and non-mathematicians alike. Perfect numbers were first studied around 300BC when Euclid first proved that if (2^p − 1) is prime then (2^(p−1))(2^p − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematicians. Throughout history, many more mathematicians played a role in bettering our understanding of perfect numbers. One of them was Ibn al-Haytham, who first suggested that the opposite (converse) of Euclid's theorem was also true: not only do numbers of the form (2^p − 1) generate perfect numbers, but all even perfect numbers are generated this way. This was eventually proven by Euler in the 1700s. A clear, animated explanation of this proof is linked here. Today, 51 perfect numbers have been discovered, with the largest having 49,724,095 digits - over three million more digits than the 50th perfect number! Even though new even perfect numbers are discovered almost every year, some questions still remain:
Infinite even perfect numbers Sometimes we see prime numbers that have the form (2^p − 1), where p is a prime number. These numbers are called Mersenne primes. As mentioned above, Euclid proved that if (2^p − 1) is a Mersenne prime, then (2^(p−1))(2^p − 1) is a perfect number. From this, we can see that the questions of the infinitude of Mersenne primes that of perfect numbers are linked. By proving the existence of infinite Mersenne primes, we can prove the existence of infinite even perfect numbers. Currently, no proof in either direction - that there are either a finite or an infinite number of Mersenne primes - exists. Mathematical consensus, based on empirical data and heuristic methods using harmonic series, is generally that there are likely to be an infinite number of Mersenne primes, and therefore even perfect numbers, but a concrete proof of this remains to be found. Odd perfect numbers Whether or not odd perfect numbers exist hasn’t been proven either. However, people have proved properties that odd perfect numbers must have, if there are any. Although the requirements for odd perfect numbers have become more demanding, they’re not contradictory and so it remains logically possible that such numbers exist. Yet, most experts believe that odd perfect numbers don’t exist. Wikipedia lists the properties that odd numbers must have; if you were to look at them, their rigidity would surprise you - I know I was! Among the many others, one property listed is that an odd perfect number must have at least 300 digits. Computers have been testing odd numbers to see it they are perfect for decades now, but have yet to find any odd perfect numbers. Although computer testing would be a valid way to disprove the conjecture that no odd perfect numbers exist, as a counterexample would disprove the claim, if there are indeed no perfect numbers, computer testing will not provide us with any real knowledge. Even if we test a vast number of odd numbers, we will not be able to conclusively state that the next odd number isn't perfect, so a concrete mathematical proof is required. Let us know in the comments if you enjoyed this discussion of perfect numbers, and what you'd like to see us talk about next!
0 Comments
Leave a Reply. |
Our AuthorsWe are high school and college students from around the world who are passionate about maths, and want to share that passion with others. Categories
All
|