• 529 is a composite number.
• Prime factorization: 529 = 23^2
• The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 529 has exactly 3 factors.
• Factors of 529: 1, 23, 529
• Factor pairs: 529 = 1 x 529 or 23 x 23
• 529 is a perfect square. √529 = 23

In the United States you can save money for college in a tax-free 529 College Savings Plan. If you use a 529 Plan, when you are 23, you could be making a good, square income. That is how I remember that 529 = 23².

529 dots can be made into a square, but is 529 an interesting number?

23 is the 9th prime number, and 529 is the 9th number with exactly 3 factors. Is that interesting?

What about the fact that 529 is palindrome 121 in BASE 22 because 1(22²) + 2(22) + 1(1) = 529?

Or that 529 can be written as the sum of three squares three ways and each of those ways contain exactly TWO elements of the set {18², 6², 3²} plus one other square number?

• 22² + 6² + 3² = 529
• 18² + 14² + 3² = 529
• 18² + 13² + 6² = 529

Wikipedia informs us that 529 is the 12th centered octagonal number because  4(12²) – 4(12) + 1 = 529.

Do any of those reasons make 529 an interesting number? Is ANY number interesting?

Recently when I wrote that 526 is a centered pentagonal number, Steve Morris of Blog, Blogger, Bloggest commented:

It’s intriguing how so many numbers have interesting or special properties. I used to think that there was something magical about this, that these patterns were somehow telling us something deep about the universe. Like, why is 3.14159… the value it is?

Nowadays I think that these special properties are things we invented. For instance, a centred pentagonal number is interesting if you think it is, and isn’t if you don’t think so. Some numbers have particularly important or curious properties (prime numbers for instance), but they are only interesting because we think they are!

What do you think?

I think that much of mathematics was discovered by astronomers, physicists, and even philosophers, and the patterns in mathematics do tell us something deep about the universe, but all of those mathematical properties existed before they were discovered. They are eternal principles that we each understand to the best of our abilities.

Are numbers with certain properties interesting only because some people think they are?

If the world never knew that pi is approximately equal to 3.14159…, we would still be as advanced as we are now. Pi seems like a very important number, but what if the world had never heard of pi? What if for thousands of years the world had instead used tau (τ ) which is approximately equal to 6.28318? Every important, magical discovery related to pi would still be known. Since τ = 2π, some people think we should all celebrate on June 28 by eating twice as much pie. Those people are really saying that tau is an interesting number only because pi is interesting. If we had never heard of pi, we wouldn’t eat pie to celebrate either day. As it is, most people have never heard of τ and would consider it to be a rather boring number.

Are any numbers inherently interesting?

Ancient mathematicians were fascinated that a series of dots could be made into different shapes. Sometimes a certain number of dots could be made into a triangle, or a square, or even a pentagon. A string of dots that couldn’t be made into a 2-dimensional rectangle represented a prime number. The fact that the ancients were interested in the shape of numbers makes them even more interesting to me.

It doesn’t matter what language you use or even what number base you use, 529 ♦’s can be arranged into a perfectly formed square just as ♦ ♦ ♦ ♦ can be. Most people can understand square numbers or even cubed numbers. However, for some people triangular numbers and pentagonal numbers are just mind-boggling, and they won’t consider such numbers to be interesting at all.

I don’t know if the ancients were aware of CENTERED pentagonal numbers, but I am fascinated by the fact that this is a shape that CAN consist of a prime number of dots (31, 181, 331, for example.)

My husband often says that nothing is boring. Two different people can listen to the same talk. One of the listeners might be moved to tears or inspired to action while the other person is bored out of his mind. It isn’t the subject matter that decides how people will react, it’s the people themselves.

Mathematician G. H. Hardy is an inspiration to me. He said, “Nothing I have ever done is of the slightest practical use.” He studied mathematical topics that interested him without regard to their usefulness. How could he potentially waste Ramanujan’s great mind on something as useless as partition theory. How did he convince Cambridge University to pay him while he obsessively explored this unimportant topic especially when he boasted that it didn’t have the slightest practical use? I don’t know how he did that, but it turns out that partition theory was not simply recreational mathematics; it does have a practical purpose! From G. H. Hardy I have learned that it is okay to explore topics that interest me even if they NEVER have a practical purpose. I haven’t figured out how to get someone to pay me to explore any unimportant topics, but I can still explore them to my heart’s content:

I often notice if the square root of a number can be reduced or not. I did not invent the idea of reducing square roots, but I’ve decided it is an interesting topic. I don’t know if anybody else looks for consecutive reducible square roots or calculates what percentage of numbers have reducible square roots, but I find it interesting, and I explore it. If you google reducible square roots, chances are that much of what you see will have been created by me.

Some mathematicians spend time trying to prove the Riemann hypothesis, but that topic doesn’t interest me as much as other topics because that is what I’ve decided.

What do you think? What makes a number interesting or not? What makes a mathematical topic interesting or not?