A Curious Coincidence

My first wife, and mother of my daughter, was born on the 23rd November 1953 and will be turning 63 shortly. My daughter was born on the same date in 1980 and will be turning 36. I've written about the mathematical aspects of this digit reversal in an earlier post. 

Getting back to my first wife however, she will be turning 23011 days old. A curious coincidence indeed. Furthermore, the number is prime and marks the start of five consecutive primes: 23011, 23017, 23021, 23027 and 23029. It is also the first member of a prime quadruple in a 2p-1 progression: 23011, 46021, 92041 and 184081. This is a Cunningham chain of the second kind. I've written about such chains in an earlier post.

However, I'm interested in the general question of what are the conditions for such coincidences to occur? It can be noted that 23011/365.242199 is about 63.00203 so such coincidences can only occur when the day count is almost exactly divisible by 365.242199. Now a person must be born between the 1st of January and the 31st December, thus the possible numbers generated (using leading zeroes) range from 01001 to 31012. 

On a spreadsheet, I tested division over every number in this range by 365.242199, looking for remainders that were less 0.01. A divergence of just two days either side produces a difference of around 0.005 so 0.01 is quite generous. A few pockets of numbers satisfied the condition but only 23011 held up in the end. 23012, corresponding to 23rd December, came close but birthdays fell on 23rd November or 23rd October so it didn't satisfy.

Even having this birthday does not guarantee the coincidence. For example, my daughter will turn 63 in 2043 but on that day she will be 23010 days old. While on the topic of coincidences, I must note that on the 23rd November 2016, I will be 24706 days old. This number factorises to 2 x 11 x 1123. Amazing.

The \(3x+1\) Problem

Today I was struggling to find something of interest to say about the daily number 24684 (see comment at end of this post) and so I considered not just its prime factorisation: \(2^2 \times 3 \times 11^2 \times17\), that consisted only of small primes, but also the factorisation \(4 \times 6171\). I wanted to find out if there was something of interest about the number 6171. It turns out that there is.

OEIS A006877 stated that in the \(3x+1\) problem, these values for the starting value set new records for number of steps to reach 1 and it listed these initial values:

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655

But what is the \(3x+1\) problem that's referred to. I'd heard of it but checked on Wikipedia to clarify my understanding. It's referred to there as the Collatz conjecture. Here is the introduction to that Wikipedia article:

The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz. The conjecture is also known as the \(3n + 1\) conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.

The conjecture can be summarized as follows. Take any positive integer \(n\). If \(n\) is even, divide it by 2 to get \(n/2\). If n is odd, multiply it by 3 and add 1 to obtain \(3n + 1\). Repeat the process (which has been called "Half Or Triple Plus One", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. 

Paul Erdős said about the Collatz conjecture: "Mathematics may not be ready for such problems." He also offered $500 for its solution. Jeffrey Lagarias in 2010 claimed that based only on known information about this problem, "this is an extraordinarily difficult problem, completely out of reach of present day mathematics."

For 6171, 261 steps are required to reach 1. At this stage, nobody has proved the Collatz conjecture. Here are some more records:

The longest progression for any initial starting number

less than 10 is 9, which has 19 steps,
less than 100 is 97, which has 118 steps,
less than 1,000 is 871, which has 178 steps,
less than 10,000 is 6,171, which has 261 steps,
less than 100,000 is 77,031, which has 350 steps,
less than 1 million is 837,799, which has 524 steps,
less than 10 million is 8,400,511, which has 685 steps,
less than 100 million is 63,728,127, which has 949 steps,
less than 1 billion is 670,617,279, which has 986 steps,
less than 10 billion is 9,780,657,631, which has 1132 steps,
less than 100 billion it is 75,128,138,247, which has 1228 steps.

See also:
https://voodooguru23.blogspot.com/2018/03/the-collatz-conjecture-revisited.html

https://voodooguru23.blogspot.com/2018/03/the-px1-map.html

https://t.co/h8cMC9QKes this link relates to Terence Tao's discovery of late 2019.

Comment regarding 24684: 

In December of 2019, I discovered that by entering a number like 24684 into the Google search box along with the acronym OEIS, a greater range of results will appear compared to just typing the number into the OEIS search bar. The reason for this is that Google will index the text files associated with the sequence. These files typically involve much larger numbers than are displayed for each sequence (possibly the first 1,000 or 10,000 rather than just the first 50).

Using 24684, as an example we find that it is a member of OEIS A228844: smallest sets of 3 consecutive abundant numbers in arithmetic progression. The initial abundant number is listed. However, only the following numbers are listed:

24, 42, 80, 100, 104, 114, 120, 126, 144, 162, 180, 196, 200, 220, 228, 234, 240, 246, 272, 282, 288, 304, 324, 348, 350, 364, 392, 402, 420, 426, 440, 460, 504, 572, 582, 588, 594, 608, 616, 624, 640, 654, 660, 666, 684, 700, 708, 714, 728, 736, 740, 786

The example given in the comments is:

24, 30, 36 is the smallest set of 3 consecutive abundant numbers in arithmetic progression so 24 is in the list.

24684 is nowhere to be found in the previous list but it does turn up in the Google search results as the 1729th member of the sequence. The arithmetic progression is 24684, 24690 and 24696 because all three are abundant with a common difference of 6 between terms.

Gaussian Integral, the Jacobian and the Gamma Function

To evaluate the Gaussian Integral requires the use of double integrals and the Jacobian for the change of variables from rectangular to polar coordinates. The following YouTube video gives a clear account of how this integral is evaluated:

The following video deals with the gamma function for value 1/2 (=1/2!) and the use of the above integral technique to evaluate it.

The final video is by the author of the second video and is an introduction to the gamma function. I'm just skimming the surface of things here but it's a start.

The Wallis Formula for Pi and the Dawson Function

I spent quite some time watching this video on the derivation of the Wallis product and practised until I could reproduce it without any external assistance. A crucial part of the solution relies on integration by parts to set up a reduction formula for the integral of \( sin(x)^n\). Here is the very well-presented and easy to understand video:

Note: the following discussion centres on integration by parts and is not related to the Wallis function.

I sometimes practise using integration by parts to solve integrals that I think of and last night my mind fell on a deceptively easy-looking integral, namely \(e^{x^2}\). The graph of this function is well-behaved and I thought that there would be an easy solution but try as I might I couldn't find it. Reluctantly, I checked first Symbolab and then WolframAlpha to find out how it could be done. Here's what the former had to say:

What on earth is this F(x) that just appears out of nowhere? WolframAlpha offered the same solution but had accompanying documentation that described F(x) as the Dawson integral defined by:

There is a quite comprehensive article about the Dawson integral or Dawson Function, as it's alternatively called, on Wikipedia but it's largely incomprehensible to me at the moment. Maybe I can come to terms with it later. 

The Harmonic Series

Here's an interesting problem: an ant traverses a circle with a circumference of one metre at a rate of one centimetre per second. After each second, the circumference of the circle increases by one metre. Will the ant ever return to its starting point?

Let's consider the matter. In the first second, the ant traverses 1% of the circumference; in the second second, it traverses 1/2%; in the third second, 1/3% and so on. The cumulative distance covered is given by the sum of:

1 + 1/2 + 1/3 + 1/4 + 1/5 + ... 

This is the harmonic series and it is divergent, meaning that the sum is continuously increasing and never reaches any upper limit. Contrast this to a convergent geometric series, the sum of whose terms approaches ever closer to 2 as more terms are added:

1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

In the case of the harmonic series, the sum will certainly surpass the 100% required for the ant to return to its starting point. It may take a long time but it will happen. In fact OEIS A004080 tells us that it would take the following number of seconds:

15092688622113788323693563264538101449859497

This of course represents more than 4.78 x 10^23 years which is far in excess of the age of the universe! The following screenshot is taken from the WolframAlpha article about the harmonic series.

Taken from http://mathworld.wolfram.com/HarmonicSeries.html

This Numberphile YouTube video was the inspiration for this post.

Reversible Digits Problem for Mother and Child

My daughter turns 36 on November 23rd and her mother, who is born on the same day, turns 63. I was struck by the reversible digits and immediately wondered whether this had happened before. A little investigation revealed that it had: at 25 and 52, at 14 and 41 and (if allowing leading zeroes) at 03 and 30. In fact it happens every 11 years. I wondered whether this was true for every mother and child. It's not: such reversibility of digits can only happen if the mother's age when she gives birth is a multiple of 9 e.g. 9, 18, 27, 36 etc. Here is my proof of this assertion. 9 is a little young of course but it has happened I think.

Let \(n\) be the age of the mother when the child is born.

Suppose mother and child turn \(x \, y\) and \(y \,x\) years old in a certain year.

Now the mother's age of \(x \, y\) has numerical value of \(10x+y\).

Similarly the child's age of \(y \, x\) has a numerical value of \(10y+x\).

We know that \( (10x+y)-(10y+x) =n\).

This means that \(9(x-y)=n\) where \(x>y\).

\(x\) and \(y\) have integer values from 0 to 9 and thus \(n\) must be a multiple of 9.

Suppose \(n=9\), this means \(x-y=1\) and there are various solutions.

Let's consider \(x=1\) and \(y=0\), giving the mother's age as 10 and the child's as 01.

Next consider \(x=2\) and \(y=1\), giving the mother's age as 21 and the child's as 12.

Next consider \(x=3\) and \(y=2\), giving the mother's age as 32 and the child's as 23.

We see can see clearly when the ages of mother and child share the same digits and why these match ups occur every 11 years.

ADDENDUM:  June 19th 2022

Here is some SageMath code that will spit out the above data (change the offset from 27 to some other number and experiment):

Engel Expansions

I've encountered Engel expansions before and today I was reminded of them again when my day count number, 24648, featured in OEIS A068379 as the Engel expansion of sinh(1/2). The initial sequence of numbers is:

1, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224

An Engel expansion is explained by Wikipedia as:

The algorithm for calculating the terms in an Engels expansion is as follows:

This is straightforward enough and I set up a worksheet in Excel to calculate the terms of the Engel expansion for whatever number I entered. I tested it out and all seemed well until I looked more closely at the terms I got for sinh(1/2). Here they are as reported by the worksheet:

The first six terms match the OEIS listing but the seventh diverges by one (623 as opposed to 624) and after that things rapidly fall apart as can be seen by comparing terms. I guess the slight errors that arise as the increasingly smaller u-th terms are divided into one quickly compound and spell disaster. Interesting illustration of the limitations of spreadsheets when very small numbers are concerned.

ADDENDUM:

It's now 1st May 2019 and I've been using SageMath for quite some time now. Here is the SageMath code to generate the Engels expansion of sinh(1/2) up to 24648 (permalink to SageMathCell):

[1, 2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648]