What is the significance of perfect numbers

We then look at the proof which is one of the few that are long enough to be nontrivial from the student view, yet easy enough to follow that they can actually understand it especially after the preliminary explorations. When it's all done, I talk a little about odd perfect numbers: no one knows of any nor can prove they don't exists.

This ties into a common thread in my upper level courses: math is still alive, still being discovered, still full of the unknown. All this being said, I surely wouldn't look down on a teacher or student who knew nothing about perfect numbers and didn't have them covered in class. I don't know of a practical application of perfect numbers per se not saying they don't exist , but its historical status dating back to the Greeks makes them interesting for pedagogical purposes.

The Greeks the Pythagoreans in particular put mystical significance on special sequences such as the primes, the figurate numbers notably the triangular numbers , and of course the perfect numbers. If you find that injecting anecdotes into your teaching helps you grab the attention of your students, then perfect numbers are a well-documented example of the ongoing human fascination with number theory since antiquity.

In fact entire MacTutor History of Mathematics Archive is quite fascinating, and makes for quite nice leisure reading. We "need" the natural numbers to count, we need the real numbers to measure and we need the complex numbers to guarantee that every quadratic equation has solutions.

Whenever we have some collection of objects and we can create a new definition that makes it possible to make distinctions between the objects some are the "same" and some are "different" the potential exists to get new insights into the objects involved, and to other related objects. Thus, we can talk about the integers that are prime, those that are squares, cubes, the sum of two squares, etc. Some of these definitions create wonderful new theoretical and applied playgrounds.

The primes are a good example. After noticing the primes then one can show the prime factorization theorem. Primes can be used to design a cryptographical system RSA powerful enough to protect many financial transactions at the current time. Perfect numbers create a "playground" for the interested.

One of my undergraduate professors, Leo Zippin, made the observation to me that if some mathematician can create a "nifty" bit of theoretical mathematics, eventually some other mathematician will find a "nifty" use for those ideas. But the question of what odd perfect numbers might be like if they exist remained open. And it remains open today. It would have to have at least nine distinct prime factors, the second-largest of which would have to be greater than 10, And it would have to have a remainder of 1 when divided by 12 or a remainder of 9 when divided by It might seem strange to prove results about numbers that might not even exist.

But every new rule narrows the search a little more. These spoofs are like generalizations of perfect numbers, and so anything true about a spoof would have to be true about a perfect number as well. Understanding odd spoofs would be especially useful, since any rule discovered for odd spoofs could be added to the existing rules for odd perfect numbers, increasing the chances of finding contradictory criteria and tightening the overall search space.

In taking up the challenge, mathematicians have broadened the concept of a spoof and have discovered a new class of numbers to study. For the most part, investigations into these spoof perfect numbers are done simply for the joy of mathematical exploration. It may seem strange to spend thousands of years hunting for numbers with curious properties, proving theorems about objects that might not even exist, and inventing new and even stranger worlds of numbers to explore.

But to a mathematician, it makes perfect sense. Are prime powers deficient or abundant? Show that N must be equal to 3. Click for Answer The smallest abundant number is The smallest odd abundant number is much, much larger! You can prove that there are infinitely many abundant numbers by proving that the multiple of an abundant number is abundant.

Of course, primes are deficient since the sum of the proper divisors of a prime is always 1. Pi, or 3. Tau, which is the But although 6. The calendar numbers of June 28th — 6 and 28 — have some very special properties that are worthy of a celebration.

Unless you were born in the year , or are a time-traveler back from the year , the only perfect numbers that will ever appear on your calendar are 6 and If you can factor a number into all of its divisors, you can immediately add them all up and discover, for yourself, whether your number is perfect or not. For the first few numbers, this is a straightforward task, and you can see that most numbers aren't perfect at all: they're either abundant or deficient. The first few countable numbers are mostly deficient, but 6 is a perfect number: the first and All prime numbers are maximally deficient, since its only factors are 1 and itself, and all powers of two 4, 8, 16, 32, etc.

The factors of the first four perfect numbers. If you exclude the numbers themselves, all the other For over a thousand years, only those first four were known.

You might look at these numbers, the ones that happen to be perfect, and start to notice a pattern here as to how these numbers can be broken down.

They're all the result of multiplying 2 to some power, let's call it X , by a prime number. And interestingly, the prime number you're multiplying it by is always equal to one less than double what 2 X is. Different ways of breaking down the first four perfect numbers reveal a suggestive pattern as to how There's a good reason for this. Remember, all powers of two — numbers like 2, 4, 8, 16, 32, etc.

At the same time, all prime numbers are maximally deficient, where their only factors are 1 and themselves. This means there are possible combinations of powers of two and prime numbers, minimally and maximally deficient numbers, that have a chance to be perfect themselves.

Not every minimally deficient and maximally deficient combination of numbers gives you a perfect number, though. Perfect Numbers List in a Table 4. How to Find a Perfect Number? Solved Examples of Perfect Numbers 6. Practice Questions on Perfect Numbers 7. Important Topics. Solved Examples Example 1 Is 28 a perfect number?

Solution: The proper factors of 28 are 1, 2, 4, 7 and Example 2 According to Euclid's proposition, if 2 p -1 is a prime number, then 2 p-1 2 p -1 is a perfect number.

Solution: The first 8 Mersenne Primes are 2, 3, 5, 7, 13, 17, 19 and Prime, p 2 p -1 2 p-1 Perfect Number, 2 p-1 2 p -1 2 3 2 6 3 7 4 28 5 31 16 7 64 13 17 19 31 Therefore, the above-mentioned are the first 8 perfect numbers. Have questions on basic mathematical concepts? Become a problem-solving champ using logic, not rules. Interactive Questions. History of Perfect Numbers.

Perfect Numbers List in a Table. Solved Examples of Perfect Numbers.