## Wednesday, 3 April 2013

### WEDNESDAY, 3 APRIL 2013

Today is the $93^{rd}$ day of the year.

$93 = 3 \times 31$

$93$ is a member of the following, two primitive Pythagorean triples:
$(93, 4324, 4325)$
$(93, 476, 485)$

$93_{10} = 1011101_{2}$ which means that $93$ is palindromic in binary. Not surprisingly there is a list of such numbers in OEIS and it is A006995.

In contrast to yesterday's Lazy Caterer number, $93$ is a Cake Number, see A000125 and Wikipedia.

The definition of Euler's Idoneal Numbers (also known as suitable, or convenient numbers) is:

The collection of the positive integers $D$ such that any integer expressible in only one way as $x^2 ± Dy^2$ (where $x^2$ is relatively prime to $Dy^2$) is a prime, prime power, or twice one of these.
Fortunately, there is an equivalent statement which says:
A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integer $a, b$, and $c$
The smallest possible values for $a, b$, and $c$ that meet the requirements of being distinct and positive are $1, 2$, and $3$.
So, $ab + bc + ac = (1 \times 2) + (2 \times 3) + (3 \times 1) = 2 + 6 + 3 = 11$.
Thus all numbers up to $11$ are Ideonal numbers.
The next smallest is the set $1, 2$, and $4$ which gives $ab + bc + ac = (1 \times 2) + (2 \times 4) + (4 \times 1) = 2 + 8 + 4 = 14$.
The next calculation is for the set $1, 2$, and $5$ which gives $ab + bc + ac = (1 \times 2) + (2 \times 5) + (5 \times 1) = 2 + 10 + 5 = 17$.
This means that we now know that the set of Ideonal numbers starts $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 15, 26$.
The Ideonal numbers become more sparse. In fact there is only one Ideonal number in the nineties and it is $93$, see A000926.