#number

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Prompt 28 - Number

@jeggyverses-jegulus-microfic May 28, Word count 322

Previous part First part

It took a lot of perseverance, but James now had Reg’s number saved in his phone. For emergencies only. It had taken a lot of work. He’d memorised Reg’s schedule and somehow found himself out in the hallway at the exact same time each day, just as Reg was leaving for work and then pulling into his parking space at the same time as Reg was driving into the parking area. Twice a day, they shared the lift, and James did his best not to talk too much, but it was hard. He hated silence and always felt the need to fill it. Reg said little, but James knew he was listening to everything he was saying as he asked whether or not he managed to fix the leaky tap in his bathroom two days after James had mentioned it. 

“Hey, Reg?” he asked one morning as they travelled down to the parking area. 

“Yes, James,” Reg sighed, though it didn’t sound as exasperated as it had when they first met. 

“I’m making chilli tonight, and I always make way too much. Would you like to come over for dinner and help me eat it?” James felt so awkward asking Reg out on a date, because that’s what it was.

Reg narrowed his eyes at him, scrutinising his entire being by the feel of it. But then he rolled his eyes and tutted. 

“Fine,” he huffed. “There had better be some of your mum’s cake though.”

James’s face broke into a huge smile, which even Reg blinked a few times; it was so bright. 

“Amazing, and yes, Mum’s bringing cake over this afternoon.”

“Good, see you this evening then,” Reg stepped out of the lift and strode across the car park to his car, leaving James still smiling in the lift as the doors closed on him, and he rode it back up to his flat and started preparing for his date. 

Next part

[a-number-a-day]

Today's number is: 5,040

(this number and fact were submitted by @hardal201)

This number is equal to 7!, so it's the number of distinct permutations of 7 objects.

It is the second number to be divisible by every positive single-digit number (so the numbers 1 through 9). The only number below 5,040 to share this property is 2,520.

It is also a highly composite number, which essentially means it has lots of divisors. Specifically, it means that it has more divisors than any smaller number.

Furthermore, it is a superior highly composite number, which means that it is highly composite relative to its size (since bigger numbers will have more divisors just by the nature of construction, we look at how many divisors it has relative to its size). In other words, it maximizes the number of divisors after penalizing larger numbers by a fixed power of their size.

5,040 has 60 divisors, which is a lot for a number that size. It's also the last factorial to be a superior highly composite number.

It is also a colossally abundant number, which again means it is unusually divisor-rich, but in a much stronger sense. Specifically, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For a given exponent ε, whichever integer has the highest ratio is called colossally abundant (for that ε). In the case of 5,040, we know that:

(sum of divisors)/n^ε is a maximum over all n

In other words, if σ(n) is defined as the sum of the divisors of n, then whichever number n maximizes:

σ(n)/n^ε

for some ε is called a colossally abundant number. Numerically, 5,040 maximizes this equation when ε is roughly around 0.23 to 0.26.

Note that this connects really closely with the famous Riemann Hypothesis. Specifically, we know that for all integers n > 5,040, Robin proved that the Riemann Hypothesis is equivalent to

σ(n) < e^γ n log log n

where γ is approximately 0.57721, the Euler-Mascheroni constant.

This inequality fails for some small n, and 5,040 is the largest exception. If the inequality holds for every n > 5,040, then the Riemann Hypothesis is true. If it fails even once above 5,040, then the Riemann Hypothesis is false.

This means that 5,040 is essentially the last safe harbor.

As a final fun fact, it was Plato's favorite number. He mentions in his dialogue Laws that 5,040 is a convenient number to use for dividing many things (including both the citizens and the land of a city-state) into lesser parts, making it an ideal number for the number of citizens (heads of families). He remarked that this number can be divided by all natural numbers from 1 to 12 with the single exception of 11. In addition, 5,038, which is two less, can be divided into 11. He also noticed that 5,040 could be divided by 12 twice over.

Plato mentioned this number so much that Benjamin Jowett, in the introduction to his translation of Laws, wrote:

"Plato, writing under Pythagorean influences, seems really to have supported that the well-being of the city depended almost as much on the number 5,040 as on justice and moderation."

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Number 4: Stealth Kragen