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A few weeks ago I found the book "Lure of the Integers" by Joe Roberts, in my schools library, and promptly ordered it from Amazon. It is a wonderful book for those of us who are interested in number theory or numbers in general. The book is a collection of integers with interesting mathematical properties. An example under the integer $10$ is: "The integer $10$ is the only composite integer such that all of its positive integer divisors other than one are of the form $a^r + 1, r > 1$."

While the "Lure of the Integers" contains many obscure and interesting integer properties, it is hardly a comprehensive collection. It may have entries for a few consecutive numbers and then skip the next six. I believe that the author was trying to limit the length of the book by including only the most interesting, obscure, or unusual, results.

Since reading the book, I've been collecting my own list of interesting integer properties.

I was hoping that some of you would share any interesting integer properties that you know of.

Two other examples that I came up with are:

$36$ is the smallest integer expressible as the sum of three distinct non-zero cubes. i.e. $36=1^3+2^3+3^3$

$5$ is the fifth number in both the Fibonacci and Perrin sequences.

I figure I will accept the answer receives the most upvotes.

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closed as too broad by Matthew Conroy, no identity, Ilmari Karonen, Magdiragdag, Thomas May 7 at 19:36

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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The interesting number paradox seems tangentially relevant. –  Jared May 7 at 18:58

3 Answers 3

There exist exactly four numbers greater than one which are the sums of the cubes of their digits, namely $$ 153=1^3+5^3+3^3,\; 370=3^3+7^3+0^3,\;371=3^3+7^3+1^3,\;407=4^3+0^3+7^3. $$

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That's interesting! Have you heard of Smith numbers? They are numbers in which the sum of their digits is equal to the sum of the digits of their prime factors. –  FofX May 7 at 17:38

$1729$ can be written as both $1^3+12^3$ and $10^3+9^3$, the smallest positive integer that can be written as the sum of two cubes (third powers) in two different ways. (If one includes negative cubes, $91$ counts; $91=6^3+(-5)^3=3^3+4^3$)

$3$ is the only number that is both a Fermat and Mersenne prime.

$2+2=2\times2=2^2$

$5^n$ and $6^n$ always end in $5$ and $6$, respectively (when written in base $10$)

$12$ is part of four different Pythagorean triples; $5, 12, 13$; $9, 12, 15$; $12, 16, 20$; and $12, 35, 37$.

In response to the property of $36$, there is a sequence related to this (sequence A031971 in the OEIS) beginning $1,5,36,354,4425,67171...$ which lists the smallest integer that can be expressed as the sum of $n$ distinct $n$th powers.

[related to another answer] Although there are no positive integers that are equal to the sum of the squares of the digits, $75$ is one greater than the sum of the squares of its digits, $7^2+5^2=74$.

$(2,4)$ is the only pair of positive integer solutions $(m,n)$ to the equation $m^n=n^m$

$6$ is the only perfect number that is also a factorial.

$49$ is the largest number whose factors (excluding itself) are all single-digit integers.

$1634=1^4+6^4+3^4+4^4$

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$1729 = 12^3 + 1^3 = 10^3 + 9^3$ –  ThePortakal May 7 at 17:37
    
Corrected now. Thanks. –  Aidan F. Pierce May 7 at 18:01
    
Correction did not take. –  Fred Kline May 7 at 18:15
    
Corrected again. –  Aidan F. Pierce May 7 at 18:18

Here is an interesting example of a larger integer, namely $$ n=196883. $$ This is the minimal dimension of a nontrivial irreducible representation of the Monster group. This is famous for unexpected connections with modular forms, and runs under the poetic name Monstrous Moonshine. This term was invented by Conway, who used it in the sense of "crazy", that $196883$ should appear in the Fourier expansion of $j(\tau)$ as the coefficient of $q$ plus $1$, i.e., $$ J(\tau)=\frac{1}{q}+744+196884q+\cdots + $$

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