2021

With 2021 almost upon us (I'm writing this on December 31st 2020), it's appropriate to look at some of the mathematical properties associated with this number. I'm quoting from Numbers Aplenty unless otherwise stated.

FIRST FACT 

2021 is rather special: it is the concatenation of two consecutive integers (20 and 21) and also the product of two consecutive primes (43 and 47). In other words:$$2021=\underbrace{2 0 2 1}_{\text{concatenate 20 and 21}}=43 \times 47$$

The next such number is:  

23073409469011482307340946901147 which is the product of the primes 4803478892324963 and 4803478892324969. 

A further example, with the two parts in increasing order like 2021, is given by the concatenation of:  

794018604377235322848433897872605582 and 794018604377235322848433897872605583.

Unsurprisingly, SageMathCell fails to factorise the catcatenated number

SECOND FACT

Let's take all the prime numbers below 100, i.e., 2, 3, 5, 7, 11, ..., 89, 97 and make domino pairs: (2, 3), (3, 5), (5, 7), ..., (83, 89), (89, 97). The sum of the numbers in all the pairs is 2021. In other words:$$2021 = \underbrace{2 + 3}+ \underbrace{3 + 5} + \underbrace{5 + 7} + ... + \underbrace{83 + 89} + \underbrace{89 + 97}$$

THIRD FACT

2021 is also equal to 33 plus the sum of the first 33 primes. In other words:$$2021 = 33 + \underbrace{2 + 3 + ... + 127 + 131 + 137}_{ \text{first 33 primes}}$$

FOURTH FACT

It is a junction number, because it is equal to \(n\)+sod(\(n\)) for \(n \)= 1996 and 2014. In other words:$$2021 = 1996 + \underbrace{25}_{\text{sod}} = 2014 + \underbrace{7}_{\text{sod}}$$

FIFTH FACT

2021 is a member of Euler's famous prime generating polynomial \(n^2+n+41\) for the case where \(n\)=44. The output in this case of course is not a prime but a semiprime. These numbers form OEIS A202018. In other words:$$2021=\underbrace{44^2+44+41}_{n^2+n+41 \text{ when }n=44}$$ SIXTH FACT 

 2021 is an emirpimes, since its reverse is a distinct semiprime:$$\underbrace{1202}_{\text{reverse }2021} = 2 \times 601$$

More on Threes and Fours

This post is a follow up on my previous post so it will make more sense if that post is read first. I made that post yesterday and today I was surprised to find that the threes and fours were still following me. Today I'm 26204 days old today and this number turns out to be divisible by four. In fact:$$26204=4 \times 6551$$So there we have a very clear connection with the number 4 but what about 3? Well, 26204 has a connection to equilateral triangles because it's a member of OEIS A171971 (see Figure 1).

A171971

Integer part of the area of an equilateral triangle with side length \(n\).

Figure 1

So an equilateral triangle with a side of 246 units has an area of 26204 square units when rounded down to the nearest whole number which closer than 99.999% of the exact value. So that's the connection with the number 3. I know that it could be argued that once you start looking for connections to 3 and 4 in the larger numbers, you'll find them but the connections of 26204 to 3 and 4 are not obscure. The connection with 4 is via its factorisation, the most fundamental characteristic of a number, and the connection to 3 is via its very close approximation to the area of an equilateral triangle with integer sides viz. 246. Of course, I could go further and point out that 246 has three digits and an average digit sum of 4 but that might be overkill.

On this same day, I visited the local McDonalds with my granddaughter. While we there, she did some drawing on her iPad and I continued reading the Pauli and Jung book referred to earlier. I had just finished reading a chapter on synchronicity when we decided to return home. Upon exiting the restaurant, I was struck by another flagrant appearance of 3 and 4 in the carpark. Figure 2 shows the sight that confronted me after descending the front stairs of the building. The signs were right in front of me and stopped me in my tracks. I got my phone out and snapped the photo shown.

Figure 2

As I was writing this post, I was reminded of my own date of birth on the 3rd April 1949 that can be written in dd/mm/yy form as 3/4/49 or mm/dd/yy form as 4/3/49. I also got thinking about my year of birth that is a \(4k+1\) prime number and thus expressible as a sum of two squares. It turns out:$$1949 = 1849+100=43^2+10^2$$Thus 3 and 4 turn up even my year of birth. I seem to be on a 3-4 roll at the moment.

The tetrahedron seems to be the perfect fusion of the 3 and 4 numbers, having four triangular faces. See Figure 3 and Figure 4.

Figure 3: source

Figure 4: tetrahedron net (source)

Prime 2-D House Numbers

I made a post on Friday, 14th February 2020, titled House Numbers that were three-dimensional figurate numbers that looked like houses (see Figure 1).

Figure 1

These numbers were characterised by two conjoined shapes: a cube, on top of which sits a square-based pyramid. The sequence of house numbers forms OEIS A051662. Today I turned 26203 days old and this turns out to be a member of another sequence, namely OEIS A229080.

A229080

Primes of the form \(T_n\) + \(S_n \) + \(1\) where \(T_n\) is the \(n\)-th triangular number and \(S_n\) is the \(n\)-th square number.

The second member of the sequence is 41 and it can be represented as shown in Figure 2:

Figure 2

It struck me that all such numbers in this sequence could be represented in this way with the square representing the body of the house, the triangle the roof and with "a cherry on top" so to speak. The "cherry" is the number 1 that must be added to triangular and square components to produce a prime (in some cases). The prime is of the form:$$n^2+\frac{n \, (n+1)}{2}+1=\frac{3n^2+n+2}{2} \text{ where } n \geq 1$$The situation with 26203 is shown in Figure 3.

Figure 3

These numbers remind me of 2-D versions of the house numbers in my earlier post and hence my choice for their description: prime 2-D house numbers. Not every sum of a square number, triangular number and 1 will produce a prime of course. In the case of \(n=2\), the total is 4 + 3 + 1 = 8 which is not prime and is the only value of \(n\) for which the sum of the square (4) and the triangle (3) is a prime.

All this of course is primary level Mathematics but part of the reason that I made this post was that the same day that had my square (4) and triangle (3) number references, I had started to read David Lindorff's "Pauli and Jung: The Meeting of Two Great Minds". In his book, the author writes:

A key to this scientific achievement was his recognition that the electrons must satisfy four quantum numbers rather than three, as had been previously assumed. It is beyond our scope to elaborate on the meaning of "quantum number" except to say that it relates to the electron's allowable energy states in an atom. The fourth quantum number was identified with what has been called an electron spin. In alchemy as well as in Jung's psychology, moving from three to four symbolises a completion, or a movement toward the centre. The alchemists identified their magnum opus with a fourfold process, which was symbolised by the so-called Axiom of Maria: one becomes two, two becomes three, and three becomes four as the one. ln association with modern dreams, Jung saw movement from three to four as symbolising a stage of inner development known as the individuation process. Pauli saw his discovery of the exclusion principle in that light.

Such a coincidence regarding threes and fours was described by Jung as synchronicity and he wrote a book titled "Synchronicity: An Acausal Connecting Principle". These types of numbers (primes that are the sum of a triangular and square number with a 1 added) are quite rare. For example, the previous such number was 21661 and the next such number will be 31177. Yet despite the rarity, this three and four related number popped up on the same day that I read the paragraph above.

Such synchronicity helps to remind me of the numinous nature of three and four and how we use such numbers daily without understanding the bottomless depths of three-ness and four-ness. The mandala is a symbol of wholeness usually encompassing in its structure the unfolding of that unity through two-fold, three-fold, four-fold division and beyond. The typical Western astrological chart is perhaps the best-known mandala and emphasises the numbers:

• 2 representing duality through:
• the dyads or signs that are diametrically opposite in the zodiac such as Aries and Libra
• the division of the chart into above the horizon and below the horizon houses
• the division of the chart into the eastern and western hemispheres
• the opposition aspect of 180°, being 360° divided by 2
• 3 representing harmony through:
• the three signs associated with each of the four triplicities of fire, earth, air and water
• the trine aspect of 120°, being 360° divided by 3
• 4 representing foundation through:
• the four signs associated with each of three quadruplicities of cardinal, fixed and mutable
• the square aspect of 90°, being 360° divided by 4
• the four sections of the chart produced by the division of the circle by the Ascendant-Descendant axis and MC-IC axis

The numbers 5, 6, 8, 9, 10 and 12 are also represented of course but I won't go into further details here about that. Figure 4 shows the triplicities and quadruplicities of the astrological chart.

Figure 4

So let's not pretend we understand these numbers and archetypal significance. We don't. We never will.

Magnanimous Numbers

I was surprised today when looking at the number associated with my diurnal age (26201) in Numbers Aplenty. It was described as a magnanimous number and I'd not heard this term before. This is not surprising as the previous such number is 24407 that occurred 1794 days ago. The source of this discovery provides the following definition:

A magnanimous number is a number (which we assume of at least 2 digits) such that the sum obtained inserting a "+" among its digit in any position gives a prime.

26201 qualifies because:$$ \begin{align}
2 + 6201 &= 6203\\
26 + 201& = 227\\
262 + 01 &= 263\\
2620 + 1 &= 2621\\
\end{align}$$and 227, 263, 2621 and 6203 are all primes.

The magnanimous numbers form OEIS A252996 and the comments on that sequence include the following:

The sequence is marked as "finite", although we do not have a rigorous proof for this, only very strong evidence (numerical and probabilistic). G. Resta has checked that up to \(5 \times 10^{16}\) the only magnanimous numbers with more than 11 digits are 5391391551358 and 97393713331910, the latter being probably the largest element of this sequence.

Numbers Aplenty includes the table shown in Figure 1: 

Figure 1

Here is the SageMath code (permalink) to generate the members of the sequence, up to 26201 (output is attached):

M=[]

for n in [11..26201]:

    OK=1

    L=list(str(n))

    for i in [1..len(L)-1]:

        start,end="",""

        for l in L[0:i]:

            start+=l

        for l in L[i:len(L)]:

            end+=l

        sum=int(start)+int(end)

        if is_prime(sum)==0:

            OK=0

    if OK==1:

        M.append(n)

print(M)

[11, 12, 14, 16, 20, 21, 23, 25, 29, 30, 32, 34, 38, 41, 43, 47, 49, 50, 52, 56, 58, 61, 65, 67, 70, 74, 76, 83, 85, 89, 92, 94, 98, 101, 110, 112, 116, 118, 130, 136, 152, 158, 170, 172, 203, 209, 221, 227, 229, 245, 265, 281, 310, 316, 334, 338, 356, 358, 370, 376, 394, 398, 401, 403, 407, 425, 443, 449, 467, 485, 512, 518, 536, 538, 554, 556, 574, 592, 598, 601, 607, 625, 647, 661, 665, 667, 683, 710, 712, 730, 736, 754, 772, 776, 790, 794, 803, 809, 821, 845, 863, 881, 889, 934, 938, 952, 958, 970, 974, 992, 994, 998, 1001, 1112, 1130, 1198, 1310, 1316, 1598, 1756, 1772, 1910, 1918, 1952, 1970, 1990, 2209, 2221, 2225, 2249, 2261, 2267, 2281, 2429, 2447, 2465, 2489, 2645, 2681, 2885, 3110, 3170, 3310, 3334, 3370, 3398, 3518, 3554, 3730, 3736, 3794, 3934, 3974, 4001, 4027, 4063, 4229, 4247, 4265, 4267, 4427, 4445, 4463, 4643, 4825, 4883, 5158, 5176, 5374, 5516, 5552, 5558, 5594, 5752, 5972, 5992, 6001, 6007, 6067, 6265, 6403, 6425, 6443, 6485, 6601, 6685, 6803, 6821, 7330, 7376, 7390, 7394, 7534, 7556, 7592, 7712, 7934, 7970, 8009, 8029, 8221, 8225, 8801, 8821, 9118, 9172, 9190, 9338, 9370, 9374, 9512, 9598, 9710, 9734, 9752, 9910, 11116, 11152, 11170, 11558, 11930, 13118, 13136, 13556, 15572, 15736, 15938, 15952, 17716, 17752, 17992, 19972, 20209, 20261, 20861, 22061, 22201, 22801, 22885, 24407, 26201]

After 26201, the next magnanimous number is 26285 and, after that, there are only three more that are less than 30000 (26881, 28285 and 28429).

More on Linear Recurrence Relations

This post is a follow on from an earlier post, titled Solving Linear Recurrence Relations of June 16th 2020, in which I dealt only with polynomials having real roots. Here I look at an example of a polynomial with one real root and two complex roots.

Suppose we have the following recurrence:$$a_n=a_{n-1}-a_{n-2}+a_{n-3} \text{ with } a_0=1, a_2=2, a_3=3$$This corresponds to \(x^3=x^2-x+1\) and thus we can write:

$$ \begin{align}
P(x)&=x^3-x^2+x-1\\
&=x^2 \,(x-1)+1 \, (x-1)\\
&=(x^2+1)(x-1)
\end{align}$$with solutions of \(x=1, \pm \, i\) when \(P(x)=0\). The \(n\)-th term can be written as a linear combination of these three solutions and thus:$$a_n=A \cdot (1)^n+B \cdot i^n+C \cdot (-i)^n \text{ with } A, B, C \text{ constants }$$We apply the initial conditions then to find the values of \(A, B, C\) by creating three equations in the three unknowns:$$\begin{align}
A+B+C&=1 \text{ when }n=0\\
A+B\,i-C \,i&=2 \text{ when }n=1\\
A-B-C&=3 \text{ when }n=2
\end{align}$$Solving these we find that:$$\begin{align}
A&=\frac{2}{5}\\
B&=-\frac{i+2}{10}\\
C&=\frac{i-2}{10}
\end{align}$$This means that \(a_n\) is then given by:$$a_n=\frac{2^{n+2}-i^{\,n}\,((i+2)-(i-2) \,(-1)^n)}{10}$$This satisfies the original conditions and will generate all terms: 0, 1, 2, 3, 6, 13, 26, 51, 102, 205, 410, 819, 1638, 3277, 6554, 13107, 26214, 52429, 104858, 209715, 419430, 838861, 1677722, ...

Even though the formula contains \(i\)'s, they always cancel out to produce real numbers. My interest in this topic was rekindled when I looked at my diurnal age (26200) and found that it could be generated by the following linear homogenous recurrence:$$a(n)=a(n-1)+a(n-3) \text{ with } a(0)=3, a(1)=1, a(2)=4 \text{ and } n≥3$$This lead to a polynomial with one real root and two complex roots, namely:$$P(x) = x^3 - x^2 - 0 \cdot x - 1 = x^3 - x^2 - 1$$However, the roots are nasty and so I chose to deal with a simpler but similar polynomial in this post. 

To see how the terms of the first sequence$$a_n=a_{n-1}-a_{n-2}+a_{n-3}$$ can be generated from the initial values$$a_0=1, a_2=2, a_3=3$$or from the formula for the \(n\)-th term$$a_n=\frac{2^{n+2}-i^{\,n}\,((i+2)-(i-2) \,(-1)^n)}{10}$$follow this SageMathCell permalink. Clearly, we also have:$$ \lim_{n \to \infty} \frac{a_{n+1}}{a_n}=2$$

Continued Fractions of Square Roots

It's all too easy, using online resources like WolframAlpha or SageMathCell, to extract a variety of information about continued fractions. For example, today I turned \(26195\) days old and this number is a member of OEIS A042431:

A042431

Denominators of continued fraction convergents to \( \sqrt{743} \).

To confirm that \(26195\) is indeed a member of this sequence, I would normally use SageMathCell but others would turn to WolframAlpha and, using the latter, it can be seen in Figure 1 that \(26195\) is indeed a denominator in one of the convergent fractions, specifically:$$ \displaystyle \frac{714024}{26195}$$

Figure 1

It's easy to take for granted that the continued fraction of an irrational number that is a square root (like \( \sqrt{743} \)) is always periodic. A good explanation of why this is so can be found on this site. In the following, I'm reproducing the site's conversion of \( \sqrt{5} \) to a continued fraction (and practising my LaTeX at the same time).

$$\begin{aligned}
\sqrt{5}&=2+x\\
5&=(2+x)^2 \\
&=4+4x+x^2\\
&=4+x\,(4+x)\\
5-4&=x\,(4+x)\\
1&=x\,(4+x)\\
x&=\frac{1}{4+x}\\
\text{ Thus } \sqrt{5}&=2+\frac{1}{4+x}\\
&=2+\frac{1}{4+\displaystyle \frac{1}{4+x}}\\
&=2+\frac{1}{4+\displaystyle \frac{1}{4+\displaystyle \frac{1}{4+x}}} \text{ etc.}\\
\end{aligned}$$Clearly, we can write \( \sqrt{5}=[2;\overline 4]\) where the overline represents repetition. In the case of \( \sqrt{743}\), we have \([27;\overline{3,1,7,27,7, 1, 3, 54}]\). The site referred to earlier goes on to develop a general algorithm for determining the periodic continued fraction of any irrational square root. The table at the end of this post lists the continued fractions of the square roots of the first 99 natural numbers (even those that aren't irrational). 

√n [ a; Period ] √1 [ 1; ] √2 [ 1; 2 ] √3 [ 1; 1, 2 ] √4 [ 2; ] √5 [ 2; 4 ] √6 [ 2; 2, 4 ] √7 [ 2; 1, 1, 1, 4 ] √8 [ 2; 1, 4 ] √9 [ 3; ] √10 [ 3; 6 ] √11 [ 3; 3, 6 ] √12 [ 3; 2, 6 ] √13 [ 3; 1, 1, 1, 1, 6 ] √14 [ 3; 1, 2, 1, 6 ] √15 [ 3; 1, 6 ] √16 [ 4; ] √17 [ 4; 8 ] √18 [ 4; 4, 8 ] √19 [ 4; 2, 1, 3, 1, 2, 8 ] √20 [ 4; 2, 8 ] √21 [ 4; 1, 1, 2, 1, 1, 8 ] √22 [ 4; 1, 2, 4, 2, 1, 8 ] √23 [ 4; 1, 3, 1, 8 ] √24 [ 4; 1, 8 ] √25 [ 5; ] √26 [ 5; 10 ] √27 [ 5; 5, 10 ] √28 [ 5; 3, 2, 3, 10 ] √29 [ 5; 2, 1, 1, 2, 10 ] √30 [ 5; 2, 10 ] √31 [ 5; 1, 1, 3, 5, 3, 1, 1, 10 ] √32 [ 5; 1, 1, 1, 10 ] √33 [ 5; 1, 2, 1, 10 ] √34 [ 5; 1, 4, 1, 10 ] √35 [ 5; 1, 10 ] √36 [ 6; ] √37 [ 6; 12 ] √38 [ 6; 6, 12 ] √39 [ 6; 4, 12 ] √40 [ 6; 3, 12 ] √41 [ 6; 2, 2, 12 ] √42 [ 6; 2, 12 ] √43 [ 6; 1, 1, 3, 1, 5, 1, 3, 1, 1, 12 ] √44 [ 6; 1, 1, 1, 2, 1, 1, 1, 12 ] √45 [ 6; 1, 2, 2, 2, 1, 12 ] √46 [ 6; 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12 ] √47 [ 6; 1, 5, 1, 12 ] √48 [ 6; 1, 12 ] √49 [ 7; ] √50 [ 7; 14 ]

√n [ a; Period ] √51 [ 7; 7, 14 ] √52 [ 7; 4, 1, 2, 1, 4, 14 ] √53 [ 7; 3, 1, 1, 3, 14 ] √54 [ 7; 2, 1, 6, 1, 2, 14 ] √55 [ 7; 2, 2, 2, 14 ] √56 [ 7; 2, 14 ] √57 [ 7; 1, 1, 4, 1, 1, 14 ] √58 [ 7; 1, 1, 1, 1, 1, 1, 14 ] √59 [ 7; 1, 2, 7, 2, 1, 14 ] √60 [ 7; 1, 2, 1, 14 ] √61 [ 7; 1, 4, 3, 1, 2, 2, 1, 3, 4, 1, 14 ] √62 [ 7; 1, 6, 1, 14 ] √63 [ 7; 1, 14 ] √64 [ 8; ] √65 [ 8; 16 ] √66 [ 8; 8, 16 ] √67 [ 8; 5, 2, 1, 1, 7, 1, 1, 2, 5, 16 ] √68 [ 8; 4, 16 ] √69 [ 8; 3, 3, 1, 4, 1, 3, 3, 16 ] √70 [ 8; 2, 1, 2, 1, 2, 16 ] √71 [ 8; 2, 2, 1, 7, 1, 2, 2, 16 ] √72 [ 8; 2, 16 ] √73 [ 8; 1, 1, 5, 5, 1, 1, 16 ] √74 [ 8; 1, 1, 1, 1, 16 ] √75 [ 8; 1, 1, 1, 16 ] √76 [ 8; 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16 ] √77 [ 8; 1, 3, 2, 3, 1, 16 ] √78 [ 8; 1, 4, 1, 16 ] √79 [ 8; 1, 7, 1, 16 ] √80 [ 8; 1, 16 ] √81 [ 9; ] √82 [ 9; 18 ] √83 [ 9; 9, 18 ] √84 [ 9; 6, 18 ] √85 [ 9; 4, 1, 1, 4, 18 ] √86 [ 9; 3, 1, 1, 1, 8, 1, 1, 1, 3, 18 ] √87 [ 9; 3, 18 ] √88 [ 9; 2, 1, 1, 1, 2, 18 ] √89 [ 9; 2, 3, 3, 2, 18 ] √90 [ 9; 2, 18 ] √91 [ 9; 1, 1, 5, 1, 5, 1, 1, 18 ] √92 [ 9; 1, 1, 2, 4, 2, 1, 1, 18 ] √93 [ 9; 1, 1, 1, 4, 6, 4, 1, 1, 1, 18 ] √94 [ 9; 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, 3, 2, 1, 18 ] √95 [ 9; 1, 2, 1, 18 ] √96 [ 9; 1, 3, 1, 18 ] √97 [ 9; 1, 5, 1, 1, 1, 1, 1, 1, 5, 1, 18 ] √98 [ 9; 1, 8, 1, 18 ] √99 [ 9; 1, 18 ]

Feynman Integration

Figure 1 shows a graphic in a tweet by Cliff Pickover that caught my attention recently:

Figure 1

It turns out that the denominator of \(x^2+1)^2\) gives the same result for \(x^2+1\) so in this video I'll just be considering the integral:$$\text{I}=\int_{-\infty} ^{\infty} \frac{\cos x}{x^2+1} \text{ d}x$$It's such an aesthetic result that I thought I'd investigate how the result was arrived at. Flammable Maths YouTube channel solves this definite integral in three different ways, although in this post I'll only be looking at the Feynman method. I'll be simply reproducing the steps outlined in the video that is shown below. It's a good opportunity to practise my LaTeX.

The first step is to introduce a variable \(t\) into the integral:$$\text{I}(t)=\int_{-\infty} ^{\infty} \frac{\cos (t \, x)}{x^2+1} \text{ d}x$$When \(t=1\), we have \(\text{I}(1)=\text{I}\). Now we'll determine the first and second derivatives. Let's start with the first derivative:$$\text{I'}(t)=\int_{-\infty} ^{\infty} \partial_t \,\frac{\cos (t \, x)}{x^2+1} \text{ d}x=\int_{-\infty} ^{\infty} \frac{1}{x^2+1} \, -x \sin (t \, x) \text{ d}x$$Now we multiply top and bottom by \(x\) and add \(0\) in the form of \(+1 + -1\):$$\text{I'}(t)=-\int_{-\infty} ^{\infty} \frac{(x^2+1-1) \, \sin (t \, x)}{x \,(x^2+1)} \text{ d}x$$We can now split the integral into two parts thanks to our \(+1 + -1\) trick:$$\text{I'}(t)=-\int_{-\infty} ^{\infty} \frac{(x^2+1) \, \sin (t \, x)}{x \,(x^2+1)} \text{ d}x+\int_{-\infty} ^{\infty} \frac{ \sin (t \, x)}{x \,(x^2+1)} \text{ d}x$$The first part of the integral simplifies to:$$-\int_{-\infty} ^{\infty} \frac{ \sin (t \, x)}{x} \text{ d}x=-\pi $$We'll just accept that result for the moment and so we have the following result for the first derivative:$$\text{I'}(t)=-\pi+\int_{-\infty} ^{\infty} \frac{ \sin (t \, x)}{x \,(x^2+1)} \text{ d}x$$Now we'll find the second derivative:$$\text{I''}(t)=\int_{-\infty} ^{\infty} \partial_t \frac{ \sin (t \, x)}{x \,(x^2+1)} \text{ d}x=\int_{-\infty} ^{\infty} \frac{ \cos (t \, x)}{(x^2+1} \text{ d}x=\text{I}(t) $$The fact the the second derivative of the function is equal to the original function means that we have a second order linear differential equation of the form:$$\text{I''}(t)-\text{I}(t)=0 \text{ where we'll assume that I}(t)=c \, e^{\, \lambda \,t}$$This means that \(\text{I''}(t)=\lambda^2 \,c\,e^{\, \lambda \, t}\) and substituting this into the differential equation we get:$$\lambda^2 \,c\,e^{\, \lambda \, t}-c \, e^{\, \lambda \,t}=0 \implies c \, e^{\, \lambda \,t}(\lambda^2-1)=0 \text{ and }\lambda=\pm1$$Thus we have: $$ \text{I}(t)=c_1 \,e^{\, t}+c_2 \,e^{-t} \text{ and }\text{I'}(t)=c_1 \,e^{\, t}-c_2 \,e^{-t}$$$$\text{I}(0)=\pi=c_1+c_2 \text{ and } \text{I'}(0)=-\pi =c_1-c_2 \implies c_1=0 \text{ and }c_2=\pi$$ $$ \text{I}(t)=\pi \,e^{-t} \text{ and thus }\text{I}(1)=\text{I} =\frac{\pi}{e}$$.Figure 2 shows a graph of the function:

Figure 2

Clearly there are negative areas under the curve that will cancel with the positive areas but the bulk of the area lies between \( \frac{-\pi}{2} \) and \(\frac{\pi}{2}\).

Here are some links that Flammable Maths provides for some of the techniques used in this video:

• Leibnitz integral rule
• Dirichlet 1
• Dirichlet 2
• Dirichlet 3

There are lots of interesting integrals like the one we have just dealt with and I am certainly out of practice in dealing with them. I should make posts like this more frequently. Looking back over my previous posts, I notice that I've made the following integration-related posts:

• The Wallis Formula for Pi and the Dawson Function (October 22nd 2016): this post deals with the following integral:$$ \int e^{x^2} \, \text{d}x$$
• Novel Integration Technique (September 29th 2017): this post also deals with Feynman's technique as it applies to the following integral:$$ \int_0^{\infty} \frac{x}{\sin x}\, \text{d}x$$
• Average Distance Between Two Points in a Square (April 28th 2019): this entails modifying an integral by changing into polar coordinates so that it is easier to deal with.

Friendly versus Solitary Numbers

Today I turned 26190 days old and discovered that this number forms one half of a friendly pair of numbers. The other half is 8148. What do these two numbers have in common? Well, we find that:$$ \frac{\sigma_1(26190)}{26190}=\frac{70560}{26190}=\frac{784}{291} \text{ and } \frac{\sigma_1(8148)}{8148}=\frac{21952}{8148}=\frac{784}{291}$$So if the sum of the divisors of one number divided by that number is the same as the sum of the divisors of another divided by that other number, then the numbers are said to be friendly. Friendly numbers are not to be confused with amicable numbers where the numbers are related in such a way that the sum of the proper divisors of one is equal to the sum of the proper divisors of the other. The smallest pair of amicable numbers is 220 and 284. 

Getting back to friendly numbers, we find that friendly triples and higher-order tuples are also possible. Friendly triples include: 

• (2160, 5400, 13104)
• (9360, 21600, 23400)
• (4320, 4680, 26208)

Friendly quadruples include: 

• (6, 28, 496, 8128)
• (3612, 11610, 63984, 70434)
• (3948, 12690, 69936, 76986)

Friendly quintuples include:

• (84, 270, 1488, 1638, 24384)
• (30, 140, 2480, 6200, 40640)
• (420, 7440, 8190, 18600, 121920)

Numbers that have friends are called friendly numbers, and numbers that do not have friends are called solitary numbers.

This ratio of the sum-of-divisors of an integer \(n\) to the integer itself is termed its abundancy and is defined as: \( \displaystyle \frac{\sigma_1(n)}{n}\).

By this definition, two numbers are friendly is they have the same abundancy.  Two numbers with the same abundancy form a friendly pair; \(n\) numbers with the same abundancy form a friendly \(n\)-tuple. 

Abundancy may also be expressed as \( \sigma _{-1}(n)\) where \( \sigma _{k} \) denotes the sum of the \(k\)-th powers of the divisors of \(n\). When \(k\)=-1, we have the sum of the reciprocals of the divisors. The abundancy of a number \(n\) should not be confused with its abundance \( A(n) \equiv  \sigma_1(n)-2n \). Refer to WolframMathWorld.

From Wikipedia we learn that:

if the numbers \(n\) and \( \sigma(n) \) are coprime – meaning that the greatest common divisor of these numbers is 1, so that \( \sigma(n)/n \) is an irreducible fraction – then the number \(n\) is solitary. For a prime number \(p\), we have \( \sigma_1(p) = p + 1\), which is co-prime with \(p\).

Thus all primes and multiples of primes are solitary. Wikipedia continues:

No general method is known for determining whether a number is "friendly" or solitary. The smallest number whose classification is unknown is 10; it is conjectured to be solitary. If it is not, its smallest friend is at least \(10^{30}\). Small numbers with a relatively large smallest friend do exist: for instance, 24 is "friendly", with its smallest friend 91,963,648.

Mutually friendly numbers as we said earlier can form friendly \(n\)-tuples that might be considered families or clubs. It's an open question whether these families have an infinite number of members. For example, it is conjectured that there are infinitely many perfect numbers but only 51 are currently known. Each perfect number has an abundancy of 2 and thus currently the perfect numbers form a 51-tuple or a family with 51 members. 

Similarly multiply perfect numbers form friendly families but firstly let's define what is meant by a multiply perfect numbers:

For a given natural number \(k\), a number \(n\) is called \(k\)-perfect (or \(k\)-fold perfect) if and only if the sum of all positive divisors of \(n\) (the divisor function, \( \sigma(n) \), is equal to \(k \times n\); a number is thus perfect if and only if it is 2-perfect. A number that is \(k\)-perfect for a certain \(k\) is called a multiply perfect number. As of 2014, \(k\)-perfect numbers are known for each value of \(k\) up to 11. Source. Also see my blog post Multiperfect, Hyperfect and Superperfect Numbers from July 24th 2019.

The club of friendly numbers with abundancy equal to 9 has 2094 known members but these multiply perfect clubs or families are thought to be finite (unlike the perfect family that is conjectured to be infinite).

There are a number of OEIS sequences associated with friendly and solitary numbers. It was stated earlier that numbers that are coprime with their sum of divisors are solitary but this is sufficient and not necessary condition for solitariness. OEIS A095739 lists those numbers that are solitary and yet not coprime with their sum of divisors:

A095739

Numbers known to be solitary but not coprime to sigma.

The first of these numbers are 18, 45, 48, 52, 136, 148, 160, 162, 176, 192, 196, 208, 232, 244, 261, 272, 292, 296, 297, 304, 320, 352, 369, ...

26190, the number that began this post, is a member of OEIS A050973:

A050973

Larger member of friendly pairs ordered by smallest maximal element.

The initial member of this sequence are:

28, 140, 200, 224, 234, 270, 308, 364, 476, 496, 496, 532, 600, 644, 672, 700, 812, 819, 868, 936, 1036, 1148, 1170, 1204, 1316, 1400, 1484, 1488, 1488, 1540, 1638, 1638, 1638, 1652, 1708, 1800, 1820, 1876, 1988, 2016, 2044, 2200, 2212, 2324, ...

The smaller members of these pairs are given by OEIS A050972:

A050972

Smaller member of friendly pairs ordered by smallest maximal element.

The initial members of this sequence are:

6, 30, 80, 40, 12, 84, 66, 78, 102, 6, 28, 114, 240, 138, 120, 150, 174, 135, 186, 864, 222, 246, 60, 258, 282, 560, 318, 84, 270, 330, 84, 270, 1488, 354, 366, 720, 390, 402, 426, 360, 438, 880, 474, 498, 510, 440, 30, 140, 534, 132, 1040, 570, 582, 606, ...

From these sequences, we can form the various pairs e.g. 28 and 6, 140 and 30 etc. Notice the two numbers (819 and 135) marked in bold in the above sequences. This pair are an example of two odd numbers being friendly. There are also cases of even being friendly to odd, such as 42 and 544635 with abundancy 16/7.

Generalised Fermat Primes

Today I turned 26188 days old and this number happens to be associated with the so-called generalised Fermat primes. Before going further, we should establish what is meant by a Fermat prime and a Fermat number. 

A Fermat number \(F_n\) is a number such that \(F_n=2^{2^n}+1\). 

If the number is prime, then we have a Fermat prime. Currently, only five such primes are known and these are:

 \(F_0=3, F_1=5, F_2=17, F_3=257, F_4=65537\)

A generalised Fermat number is a number of the form \(a^{2^n}+1\) where \(a>2\). Only if \(a\) is even can a generalised Fermat number be prime. Now 1024=\(2^{10}\) and so if we look at numbers of the form \(a^{2^{10}}+1\), we find that the values of \(a\) that produce primes are (up to 26188): 

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188

These numbers form OEIS A057002. Figure 1 shows a plot of these same numbers:

Figure 1

Many of the largest known prime numbers are generalised Fermat numbers. To date (14th December 2020), the largest such prime is:$$1059094^{2^{20}}+1=1059094^{1048576}+1 \text{ which contains } 6317602 \text{ digits }$$This prime was discovered in 2018 but it pales in comparison to a number of larger Mersenne primes, the largest of which (discovered also in 2018) is:$$2^{82589933-1} \text{ which contains } 24862048 \text{ digits }$$A list of the current largest 100 primes can be found here.

It should be noted that there is another less common definition of a generalised Fermat number and that is:$$F_m(a,b)=a^{2m}+b^{2m} \text{ with gcd(\(a,b\))=1}$$I've looked at generalisations or extensions of other number types in the past, specifically:

• Generalised Cunningham Chains
• Fibonacci-like Sequences
• Beyond Fibonacci
• Root-Mean-Square and other Means

UPDATE on Thursday, February 4th 2021

Today I turned 26240 days old and one of the properties of this number, as with 26188 that is dealt with in this post, is that it is a generalised Fermat prime. Furthermore, both numbers belong to OEIS A057002:

A057002

Numbers n such that n^1024 + 1 is prime (a generalized Fermat prime).

In fact, looking at the members of the sequence, we see that there is a cluster of three numbers (26132, 26188, 26240) and Figure 2 makes this even more apparent:

1, 824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, 6640, 7026, 7158, 9070, 12202, 12548, 12994, 13042, 15358, 17646, 17670, 18336, 19564, 20624, 22500, 24126, 26132, 26188, 26240, 29074, 29658, 30778, 31126, 32244, 33044, 34016, ...

Figure 2: cluster of generalised Fermat primes