## 1 is the Only Square-free Square

How can a square be square-free? In order for square-freeness to be interesting, it must be, and is, defined in terms of divisibility by **non-trivial** squares. So to create this particular mathematical oxymoron, one just needs a trivial square, namely 1.

I collect exciting properties of integers on my Number Gossip website. Did you know that forty is the only number whose letters appear in alphabetical order when written in English? Or that the largest amount of money one can have in US coins without being able to make change for a dollar is 119 cents?

Recently I wrote about a weird occasion that motivated me to search for new properties. Here is a sample of some amusing new updates.

- 68 is the last 2-digit string to appear in the decimal expansion of pi.
- 77 is the smallest number n such that the smallest possible number of multiplications required to compute x to the n-th power is by 1 fewer than the number of multiplications obtained by Knuth’s power tree method.
- 195 is the smallest groupless number; that is, it is the smallest number n whose set of pairwise products up to n cannot be completed to the multiplication table of a finite group of order n (submitted by Andrew Pollington).
- Digits in the Morse code can be represented in base 3 with 1 for a dot and 2 for a dash; Morse-coded zero in base 3 is evaluated to 242.
- 247 is the smallest possible difference between two integers that together contain each digit exactly once.
- An equilateral triangle, whose area and perimeter are equal, has an area (and perimeter) equal to the square root of 432.
- 480 is the smallest number such that when written in hexadecimal (1E0), looks like another number in scientific E notation.
- 510 is the smallest number that is not a palindrome, even after removing trailing zeros, which is divisible by its reversal. It is trivial that palindromes with trailing zeros are divisible by their reversal, making this the first interesting case.
- 960 is the number of different starting positions in the Fischer random chess game. The Fischer random chess was invented, not surprisingly, by Bobby Fischer. The starting position is created like this: White’s non-pawn pieces are placed randomly on the first row so that the two bishops are on different colored squares, and the king is between the rooks to allow to castle. The white pawns are placed on the second row as usual, and the black pieces are placed symmetrically with respect to the horizontal line symmetry.
- 1000 is the smallest number whose scientific notation (1E3) is shorter than its decimal representation.
- 1084 is the smallest number whose American English name contains all five vowels in order.
- 1642 is the smallest number n without zeros so that for each digit d of n, the number 2
^{d}is a substring of n. - 1659 is the smallest number n such that its Roman numeral occupies the n-th position when all Roman numerals are sorted in the lexicographic order.
- 2187 is the smallest compact number: numbers that can be expressed more compactly using their prime factorization than their decimal expansion, where multiplication and power contribute one character (2187 is written as 3^7).
- 2500 is the smallest number n whose distinct prime factors are exactly the same as the distinct prime factors of each of the numbers obtained by deleting any single digit of n.
- 8542 is the largest integer that can’t be represented as a sum of squares of numbers whose reciprocals sum to 1. This is a recent (2018) result by Max Alekseyev.
- 8719 is the smallest number n such that the smallest possible number of multiplications required to compute x to the n-th power is by 2 fewer than the number of multiplications obtained by Knuth’s power tree method.

## Graham:

All fun results but I will admit to not really enjoying the ones which are dependent on decimal representation.

It would be interesting to divide all your properties into 2 classes: those which are true in all representations, and those which are dependent on decimal (or some other) representation. The former feel like properties of the number, the latter more as properties of the string of digits.

Of course, the REALLY interesting case would be to find numbers with properties which are about the representation but held for ALL bases. For example, if the property of 68 held in all bases: something like, 68 is the largest number N such that N, represented in any base B (taking m digits) is the last m-digit string of digits in the base-B representation of pi, for all B.

Of course, then I would ask you whether the property held in bases which are not successive powers of a single number (that has been an interest of mine since I found out, in my first job in 1978, that APL included a simple operator that could convert a value to a vector of numbers representing the value in a base formed from arbitrary numbers – it was designed for things like converting weights to stones, pounds and ounces but seemed like an interesting area for research).

17 August 2021, 5:26 am## L33tminion:

> 68 is the last 2-digit string to appear in the decimal expansion of pi

I misread this as some sort of nonsense joke about the “last 2-digit string to appear in pi” before realizing you meant “68 is the two-digit string that initially appears in the decimal expansion of pi farthest from the start”.

20 August 2021, 12:26 pm## tanyakh:

L33tminion, thanks. It does sound confusing. I will change it to:

20 August 2021, 3:15 pmOut of all 2-digit strings, 68 is the last to appear in the decimal expansion of pi.