Greek Letters

Given the importance and prevalence of Greek letters in Mathematics, I thought it high time that I make a post to summarise information on this topic. Here is a list of Greek letters and their names taken from Wikipedia:

Click image to see it more clearly

In symbolab (a recent discovery described in a post to my Pedagogical Posturing blog), these are the ways the letters appear:

Some mathematical functions that use these letters and that I'm familiar with or at least heard about are:

The screenshot above was taken from my versal site and here is a list of the ASCIIMathML that was used to create it:

Click image to see it more clearly

Lycrel Numbers

I've referred to the Lychrel numbers before in a couple of earlier posts but they always keep cropping up and a dedicated post will serve to remind of what they are, specifically a set of numbers that do not form a palindrome through the process of reversing and adding their digits. Of course, in base 10 it hasn't been proved that such numbers do not form palindromes somewhere down the iterative track but the first Lychrel number, 196, has been tested to a billion digits and no palindrome has been found. There is a site dedicated to these numbers: http://www.p196.org although it hasn't been updated in many years.

Wikipedia says that "about 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps. About 90% resolve in seven steps or fewer". The article goes to note that "89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188" and "10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps". These statistics are relevant because the number of the day when I'm composing this post - 24636 - is a member of OEIS A065320: 53 'Reverse and Add' steps are needed to reach a palindrome. 

The first numbers in this sequence are:

10677, 11667, 12657, 13647, 14637, 15627, 16617, 17607, 20676, 21666, 22656, 23646, 24636, 25626, 26616, 27606, 30675, 31665, 32655, 33645, 34635, 35625, 36615, 37605, 40674, 41664, 42654, 43644, 44634, 45624, 46614, 47604, 50673

The various milestones when a number sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome are recorded in OEIS A065198. The first few such numbers are 0, 10, 19, 59, 69, 79, 89, 10548, 10677, 10833, 10911, 147996, 150296.

More information can be found on this site: https://www.dcode.fr/lychrel-number. A number is delayed when there a multiple steps before becoming a palindrome. The most delayed known is 1186060307891929990 with 261 iterations. 

There are potential Lychrel primes and the first three of these are 691, 887 and 1997. These primes form OEIS A135316:

A135316

Primes in A023108(n); or Lychrel primes.

Here is a list of the initial members:

691, 887, 1997, 3583, 3673, 3853, 3943, 4079, 4259, 4349, 4799, 4889, 5581, 5851, 6257, 6977, 8089, 8179, 8269, 8539, 8629, 8719, 10663, 10883, 11777, 11833, 11867, 11923, 11953, 11959, 12097, 12763, 12823, 13397, 13523, 13553, 13597, 13633

on June 6th 2021

Unusual Function

While investigating the problem of finding the ratio between radius and height that provides the minimum surface area for a fixed volume, I made a mistake and ended up with what turned out to be an usual function. Stripped of such extraneous constants, this is the function that I discovered:

The unusual behaviour that it has is that it is discontinuous and undefined at x=0 and yet it has limits of 1 and -1 respectively as x --> 0 from the positive and negative directions. Here is what WolframAlpha turned up:

Furthermore, my mobile version of WolframAlpha shows the step-by-step solutions to the derivation of the limit:

So there we have it and I obviously have to brush up on l'Hôpital's rule which I certainly remember from the old days but the details completely elude me now.

Semiprime Factor Ratios

All biprimes (or semiprimes or 2-almost-primes) can be visualised as unique rectangles and all triprimes (or 3-almost-primes) as rectangular prisms. I only intend to deal with biprimes in this post. Let's take a recent biprime, 24581 = 47 x 523, as a starting point. It can be visualised as a rectangle with a width of 47 units and a length of 523 units. It's the ratio of width to length that's of interest. 

A golden semiprime is defined as a number that factors to: 

• \(p \times q\) (with \(p<q\)) and
• \( |p \times \phi-q|<1 \), where \( \phi\) is the golden ratio of \( \dfrac{1+\sqrt 5}{2} \)

Clearly 24581 does not satisfy this condition and not many semiprimes do. The next for me is 27641 which factors to 131 × 211 and where:$$|131\times \phi-211| \approx 0.9624525$$and so it just barely satisfies the criterion. Here is a partial list as shown in OEIS A108540:

6, 15, 77, 187, 589, 851, 1363, 2183, 2747, 7303, 10033, 15229, 16463, 17201, 18511, 27641, 35909, 42869, 45257, 53033, 60409, 83309, 93749, 118969, 124373, 129331, 156433, 201563, 217631, 232327, 237077, 255271, 270349, 283663, 303533, 326423

Presumably there is an infinity of golden semiprimes. There are other ratios of interest, for example pi. Here the number 154 = 7 x 22 could be treated in a manner similar to the golden semiprimes and the question asked as to whether \( |7 \times \pi-22| \) is less than 1. It turns out that it is (0.9911...) and so could perhaps be termed a circular semiprime. The number 15883 = 71 x 223 yields a much closer result (0.053...). Similarly for \(e\), the number 133 = 7 x 19 yields \( |7 \times e - 19| \approx 0.02797 \) and could be termed an Euler semiprime for want of a better term. 

Some semiprimes are not related to special mathematical constants but are nonetheless of interest. For instance, for Friday 26th August 2016 (the day I'm completing this post), my number 24617 = 239 x 103 and the ratio 239:103 can be expressed approximately as 2.32:1 (rounding off 2.320388... to two decimal places). This is very close to the aspect ratio for the current widescreen cinema standard of 2.35:1 or 2.39:1. However, following the pattern for the golden semiprime ratio, the result of \( |103 \times 2.35-239|=3.05 \) and \( |103 \times 2.39-239|=7.17 \) mean that the results are outside the acceptable range (less than 1).

Another way to view the ratio 239:103 is as 0.69883:0.30117 and if we round off to two decimal places, the result is 0.70:0.30 or 70% : 30%. This is the ratio of copper to zinc in so-called Cartridge brass described as follows:

70/30 brass has excellent ductility and good strength. It is often used where its deep drawing qualities are needed. The alloy is the most common brass in sheet form (source).

I guess the concept of the golden semiprime has opened my eyes to other classifications of semiprimes based on other constants such \(e\) and \( \pi\). Expressing the ratio in such a way that both sides sum to 1 is also useful because, as in the case of 0.70:0.30, connections to physical applications can be drawn.

on August 30th 2021

Cuban Primes

Investigation of today's prime day, 24571, revealed that it is a cuban prime, a prime that is a difference of two consecutive cubes (OEIS A002407). In the case of 24571, the two cubes are \(91^3\) and \(90^3\). A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y:

• the first is \(\displaystyle \frac{x^3-y^3}{x-y} \) where \(x=y+1\) and \(y>0\)
• the second is \( \displaystyle \frac{x^3-y^3}{x-y}\) where \(x=y+2\) and \(y>0\)

24571 obviously belongs to the first type with \(y\) = 90 and \(x\) = 91. To quote from Wikipedia: "the name "cuban prime" has to do with the role cubes (third powers) play in the equations, and has nothing to do with Cuba". 

Returning to the first equation, it can be seen that it simplifies to \( (y+1)^3-y^3 \) and then to \( 3y^2+3y+1 \) which is the equation of centred hexagonal numbers. Thus every cuban prime of the first type is a centred hexagonal number. These latter numbers begin 1, 7, 19, 37, 61, 91, 127, 169, ... etc. Many of the centred hexagonal numbers are prime because they all end in 1, 7 or 9.

Because 24571 is more distant than usual from its neighbouring primes, it also gains entry into the OEIS via A137875: prime numbers, isolated from neighbouring primes by more than 16 and A163111: prime numbers with gaps larger than 18 towards both neighbouring primes. The nearest primes are 24551 and 24593.

There are also quartan, quintan and sextan primes. For example quintan primes can be of the form:$$\frac{x^5-y^5}{x-y} \text{ or } \frac{x^5+y^5}{x+y}$$An example of a quintan prime is \( 4651 = \displaystyle \frac{6^5-5^5}{6-5}\). 

Another example is \( 26321 = \displaystyle \frac{11^5-5^5}{11-5} \).

It should be noted that \( x-y \) will always divide \(x^n-y^n\) for \(n \geq 1 \) and in the case of \(x^5-y^5\) this division yields \(x^4 + yx^3 + y^2x^2 + y^3x + y^4\). Thus the situation of \(x=y\) is possible and in the case of \(x=y=1\) yields the quintan prime 5. Primes of this form make up OEIS A002649. 

A002649

Quintan primes: \(p = \dfrac{x^5 - y^5 }{x - y} \)

If we return to the definition of a cuban prime as being the difference of two consecutive cubes, then an equivalent definition of a quintan prime as being the different of two consecutive fifth powers does not necessarily hold. It does in the case of 4651 where 5 and 6 differ by 1 but in the case of 26321 the difference between 5 and 11 is 6. This makes for some degree of confusion which I might try to clarify further at some future date.

on April 27th 2021 and

on October 4th 2023

Pythagorean Numbers

Of course I knew about Pythagorean triples and even primitive Pythagorean triples but I hadn't heard of Pythagorean numbers. The term emerged when I was researching my daily number, 24570, using the OEIS. This number was paired with 24576 and the smaller followed the larger in sequence A228875: Pairs of Pythagorean numbers differing by 6. This difference is apparently the minimum possible. The sequence started:

24, 30, 54, 60, 210, 216, 330, 336, 480, 486, 540, 546, 720, 726, 750, 756, 1344, 1350, 1710, 1716, 2160, 2166, 8664, 8670, 8970, 8976, 10080, 10086, 10290, 10296, 12144, 12150, 15600, 15606, 18144, 18150, 24570, 24576, 28560, 28566, 30240, 30246, 34650, 34656

This didn't really explain what constituted a Pythagorean number. However, as I discovered here, the definition of such as number is that it is the area of a Pythagorean triangle and primitive Pythagorean number is the area of a primitive Pythagorean triangle. Sequence A009111 provides an ordered list the areas of Pythagorean triangles, effectively providing a list of the initial Pythagorean numbers. Oddly, 24570 turns out to be 294th and 295th in this list. The reason for this will soon become clear.

While I knew that 24570 was a Pythagorean number and thus the area of a Pythagorean triangle, I didn't know the integer sides that comprised such a triangle but it seemed that there were two possible triangles because the number occupied two positions in the list. It took a little fiddling around in WolframAlpha to come up with the numbers.

Thus the triangles were 84, 585, 591 and 180, 273, 327. The number 24570 is not a primitive Pythagorean number because the members of each triplet are divisible by three. The equivalent Pythagorean triplets are 28, 195, 197 and 60, 91, 109.

The Digits of Pi

Today's numbered day is 24566 and a check with the OEIS showed its connection to the digits of \(\pi\). Specifically, OEIS A083625 records the starting positions of strings of three 6's in the decimal expansion of \(\pi\). The first elements in the sequence are as shown below:

2440, 3151, 4000, 4435, 5403, 6840, 10163, 10335, 10591, 13594, 15888, 16109, 18504, 20231, 21880, 21881, 23057, 23511, 24566, 25948, 26212, 27703, 27841, 29666, 29868, 29869, 32427, 32428, 33363, 36353, 38132, 40370, 40650, 43523

Wolfram Mathworld has collected some interesting information about peculiarities in the digits of \(\pi\). For a start, OEIS A050285 lists the starting position of the first occurrence of a string of \(n\) 6's in the decimal expansion of \(\pi\), starting with \(n\)=1. The initial terms are 7, 117, 2440, 21880, 48439, 252499, 8209165, 45681781, 45681781, 386980412. It can be seen that \(n\)=3, corresponding to 666, occurs initially at position 2440. 6666 (\(n\)=4) occurs at position 21880 and this is reflected in OEIS A083625 which shows 666 at 21880 and 21881.

Many OEIS sequences relate to the digits of \(\pi\). Here are some of them:

• starting positions where 0123456789 occurs (OEIS A101815)
  
  
• starting positions where 9876543210 occurs (OEIS A101816)
  
  
• starting positions of the first occurrence of \(n\)=0, 1, 2, ... in the decimal expansion of \(\pi\) (including the initial 3 and counting it as the first digit) are 33, 2, 7, 1, 3, 5, 8, 14, ... (OEIS A032445)
  
  
• \(\pi\)-primes, i.e., \(\pi\)-constant primes occur at 2, 6, 38, 16208, 47577, 78073, ... (OEIS A060421)
  
  
• starting positions for repeating digits e.g. 6666 occurs at 21880

0 - A050279: 32, 307, 601, 13390, 17534, 1699927, ... 

1 - A035117: 1, 94, 153, 12700, 32788, 255945, ...

2 - A050281: 6, 135, 1735, 4902, 65260, 963024, ...

3 - A050282: 9, 24, 1698, 28467, 28467, 710100, ...

4 - A050283: 2, 59, 2707, 54525, 808650, 828499, ...

5 - A050284: 4, 130, 177, 24466, 24466, 244453, ...

6 - A050285: 7, 117, 2440, 21880, 48439, 252499, ...

7 - A050286: 13, 559, 1589, 1589, 162248, 399579, ... 

8 - A050287: 11, 34, 4751, 4751, 213245, 222299, ... 

9 - A048940: 5, 44, 762, 762, 762, 762, 1722776, ...

999999 occurs at position 762 and is known as the Feynman point.

on January 7th 2021