## 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.

## Tuesday, 2 April 2013

### TUESDAY, 2 APRIL 2013

Today is the $92^{nd}$ day of the year.

$92 = 2^2 \times 23$

$92$ is a Lazy Caterer number, see A000124. It is the maximum number of pieces of a pizza (not all the same size) that can be created with $13$ cuts.

$92$ is a member of the following, two primitive Pythagorean triples:
$(525, 92, 533)$
$(2115, 92, 2117)$

There $92$ ways of placing $8$ queens on a standard chess board so that none of the queens are under attack from another queen, see A000170.

$92$ is a Pentagonal number, see A049452.

## Monday, 1 April 2013

### MONDAY, 1 APRIL 2013

Today is the $91^{st}$ day of the year and since $91 \approx \frac {365} {4}$ then we are one quarter of the way through the year.

$91 = 7 \times 13$

$91 = 1 + 2 + 3 + 4 + 5 + 6 + 7= 8 + 9 + 10 + 11 + 12 + 13$ which means that $91$ is a triangular number, see A000217.

$91 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 1 + 4 + 9 + 16 + 25 + 36$ which means that $91$ is a square pyramidal number, see A000330.

$91$ is a member of the following two primitive, Pythagorean triples:
$(602, 912, 1092)$
$(912, 41402, 41412)$

Repeatedly, square the digits of a number and sum them.
$9^2 + 1^2 = 81 + 1 = 82$
$8^2 + 2^2 = 64 + 4 = 68$
$6^2 + 8^2 = 36 + 64 = 100$
$1^2 + 0^2 + 0^2 = 1 + 0 + 0 = 1$
Since the result is the number $1$ then $91$ is, by definition, a Happy Number, see A007770.

$\frac {91! + 2} {2} = \frac {135,200,152,767,840,296,255,166,568,759,495,142,147,586,866,476,906,677,791,741,734, 597,153,670,771,559,994,765,685,283,954,750,449,427,751,168,336,768,008,192,000,000, 000,000,000,000,000 + 2}{2} = \frac {135,200,152,767,840,296,255,166,568,759,495,142,147,586,866,476,906,677,791,741,734, 597,153,670,771,559,994,765,685,283,954,750,449,427,751,168,336,768,008,192,000,000, 000,000,000,000,002}{2} = 67,600,076,383,920,148,127,583,284,379,747,571,073,793,433,238,453,338,895,870,867, 298,576,835,385,779,997,382,842,641,977,375,224,713,875,584,168,384,004,096,000,000, 000,000,000,000,001 \approx. 6.76 \times 10^{139}$ and that is prime, see A082672.
In much the same way as one might be amazed that the invention of bread making ever came about then I cannot fail to be amazed that
1) We know that this 140 digit number is prime and
2) We know that this 140 digit number is $\frac {91! + 2}{2}$ and
3) The two facts have been associated with each other.