A Plethora Of Squares

The number associated with my diurnal age today, 27625, has the unique quality that it is the smallest number capable of being expressed as a sum of two squares in exactly eight different ways. Here are the different ways:$$ \begin{align} 20^2+ 165^2 &= 27625\\27^2+ 164^2 &= 27625\\45^2+ 160^2 &=27625\\60^2+ 155^2 &=27625\\83^2+ 144^2 &=27625\\88^2+ 141^2 &= 27625\\101^2+ 132^2 &=27625\\115^2+ 120^2 &=27625 \end{align}$$The number arises from 27625's factorisation where:$$27625 = 5^3 \times 13 \times 17$$To determine the number of ways in which it can be written as the sum of two squares, we add 1 to each index, multiply them together and divide the product by 2. If the product is not even, then we round the result up. In the case of 27625 we have:$$ \begin{align} \frac{(3 +1) \times (1 +1) \times (1+1)}{2} &= \frac{4 \times 2 \times 2}{2} \\ &=8 \end{align}$$This property of 27625 qualifies it for membership in OEIS A016032:

 A016032: least positive integer that is the sum of two squares of positive integers in exactly \(n\) ways.

The initial members of the sequence are:

2, 50, 325, 1105, 8125, 5525, 105625, 27625, 71825, 138125, 5281250, 160225, 1221025, 2442050, 1795625, 801125, 446265625, 2082925, 41259765625, 4005625, 44890625, 30525625, 61051250, 5928325, 303460625, 53955078125, 35409725, 100140625, 1289367675781250

It can be noted that the sequence is not monotonic increasing. For example, 8125 is the smallest number that can be expressed as a sum of two squares in exactly five ways but 5525 is the smallest number that can be expressed as a sum of two squares in exactly six ways.

Now if we square 27625 we have the following factorisation:$$27625^2=5^6 \times 13^2 \times 17^2$$Now this number can be expressed as a sum of two squares in 32 different ways. These are the possible ways:$$ \begin{align} 0^2+ 27625^2 &= 27625^2\\969^2+ 27608^2 &=27625^2\\1175^2+ 27600^2 &=27625^2\\2625^2+ 27500^2 &=27625^2\\3060^2 +27455^2 &=27625^2\\ 3588^2+ 27391^2 &=27625^2\\ 4225^2+27300^2 &=27625^2\\5180^2+ 27135^2 &=27625^2\\5655^2+ 27040^2 &=27625^2\\6600^2+ 26825^2 &=27625^2\\6800^2+ 26775^2 &=27625^2\\7223^2+ 26664^2 &=27625^2\\7735^2+ 26520^2 &=27625^2\\8856^2+ 26167^2 &=27625^2\\9724^2+ 25857^2 &=27625^2\\10220^2+ 25665^2 &=27625^2\\10625^2+ 25500^2 &=27625^2\\10815^2+ 25420^2 &=27625^2\\11700^2+ 25025^2 &=27625^2\\12137^2+ 24816^2 &=27625^2\\13000^2+ 24375^2 &=27625^2\\13847^2+ 23904^2 &=27625^2\\14025^2+ 23800^2 &=27625^2\\14400^2+ 23575^2 &=27625^2\\15620^2 +22785^2 &=27625^2\\16575^2+ 22100^2 &=27625^2\\17340^2+ 21505^2 &=27625^2\\17500^2+ 21375^2 &=27625^2\\18239^2+ 20748^2 &=27625^2\\18600^2+ 20425^2 &=27625^2\\18921^2+ 20128^2 &=27625^2\\19305^2+ 19760^2 &=27625^2 \end{align}$$Once again, we know that there are 32 ways to write this number as a sum of two squares because looking at the indices again we have:$$\begin{align} \frac{(6+1) \times (2+1) \times (2+1)}{2} &= \frac{7 \times 3 \times 3}{2}\\ &= \frac{63}{2}\\ &\rightarrow 32 \text{ rounded up} \end{align}$$Now this property of the square of 27625 qualifies it for membership in OEIS A097244:

A097244: numbers \(n\) that are the hypotenuse of exactly 31 distinct integer-sided right triangles, i.e., \(n^2\) can be written as a sum of two squares in 31 ways.

By 31 ways and not 32 ways is meant that the number can be written a sum of two distinct non-zero numbers in 31 ways. The initial members of this sequence are:

27625, 47125, 55250, 60125, 61625, 66625, 78625, 82875, 86125, 87125, 94250, 99125, 110500, 112625, 118625, 120250, 123250, 129625, 133250, 134125, 141375, 144625, 148625, 155125, 157250, 157625, 164125, 165750, 172250, 174250, 177125

As can be seen, 27625 is the first member of this sequence. 

Perimeters and Pythagorean Triples

My diurnal age today is 26800 and one of the properties of this number is that it's a member of OEIS A010814: perimeters of integer-sided right triangles. Such numbers are relatively frequent (almost 22%) as can be seen in the data below:

5824th member is 26780

5825th member is 26784

5826th member is 26790

5827th member is 26796

5828th member is 26800

5829th member is 26808

5830th member is 26810

5831th member is 26814

5832th member is 26820

As I discovered however, it's not easy to find the sides of the triangles with these perimeters. The \(m,n\) method for generating primitive Pythagorean triples is shown in Figure 1:

Figure 1

Thus we have the shortest side \(a=m^2-n^2\) with \(m \gt n\) with \(b=2mn\) and the hypotenuse \(c=m^2+n^2\) where \(m\) and \(n\) are integers. Using these formulae in SageMathCell to generate the triples and associated perimeters, I found that the calculation timed out once the perimeters got even moderately large (far smaller than 26800 for example). Fortunately, there was a web resource that I could call upon. See Figure 2:

Figure 2: https://r-knott.surrey.ac.uk/Pythag/pythag.html

As can be seen, the calculator spits out a plethora of data including the perimeter. It turns out that there are three different right angled triangles with a perimeter of 26800: (6800, 8844, 11156), (5360, 10050, 11390) and (4355, 10800, 11645). If the triangle's dimensions are a multiple of those of a primitive triangle then this is shown as well. This website is certainly the ultimate resource for information about right-angled triangles.

Sometimes there is only one right-angled triangle with a given perimeter. For example, the one triangle with perimeter 26814 has dimensions (3052, 11685, 12077) and it is primitive.  Perimeters that arise from only one triangle form OEIS A098714: only one Pythagorean triangle of this perimeter exists. 26814 is the 2944th member of this sequence and so such numbers make up slightly under 11% of all numbers (at least in the range up to 26814).

The subsite referenced in Figure 2 is maintained by a Dr. Ron Knott and the main site contains links to many other interesting mathematical concepts. Here is some biographical information:

Ph.D (1980, University of Nottingham), M.Sc (1976, University of Nottingham), B.Sc (Pure Maths, University of Wales), C.Math, FIMA, C.Eng, MBCS, CITP

Visiting Fellow, Department of Mathematics,

formerly Lecturer in Mathematics and Computing Science Departments (1979-1998)

Faculty of Electronics and Physical Sciences,

University of Surrey,

Contact me initially by Email: ronknott at mac dot com

I was a lecturer in the Departments of Mathematics and Computing Science at the University of Surrey, Guildford, UK, for 19 years until September 1998 when I left to start working for myself making web pages for maths education sites.

I now give mathematics talks to students at schools and universities as well as to general audiences, teachers' conferences and Science Festivals on topics of the web pages above, especially the Fibonacci Numbers and why they occur so often in plants, Fun with Fractions, As Easy As Pi etc.

I now live in Bolton, near Manchester in NW England.

Dream Numbers

This post is a little different from my usual content that is often prompted by an analysis of the number associated with my diurnal age. This post is prompted by a dream that I had involving a shipping container that, unlike most such containers, was gun metal in colour. It seemed totally sealed but I found a tiny opening and lit a match (or used the torch on my phone) to peer within. I saw a woman peering back at me.

She invited me in and the front face of the container disappeared so that access was now possible. She said that there were 37 people inside. Two of those were adolescent boys, one noticeably shorter than the other. It seemed that one of them was 17 but I wasn't sure which one. I pointed to the taller boy and then, quite audibly in my dream, said "seventeen?". The woman, and the mother of the two boys, smiled and clarified the situation. She said that they were twins (clearly not identical) and had been born on a Saturday. They were both 17.

After some thinking about these numbers, I realised that both 17 and 37 are 4\(k\)+1 primes and thus form the hypotenuse of right angled triangles with associated integer sides, connected by Pythagoras' Theorem:$$ \begin{align} 17^2&=8^2+15^2\\37^2&=12^2+35^2 \end{align}$$Thus we end up with a set of six numbers:$$8, 12, 15, 17, 35, 37$$Being 4\( k\)+1 primes of course, we can also write:$$ \begin{align} 17&=1^2+4^2\\37&=1^2+6^2 \end{align}$$This generates a set of five numbers:$$1, 4, 6, 17, 37$$I tend to go with the set of six numbers as they are easily associated with the Saturday night draw in the Australian lottery system. 

Being in Indonesia, I can't participate in this lottery, not even using a VPN, so I passed these numbers on to my daughter-in-law who lives in Melbourne. I suggested that she try them out. Whether she does or doesn't, I'll report back on what numbers came up in the Saturday night draw at the end of this post. I won't make the post public until after Saturday night in case anyone tries to "cash in" on my dream numbers!

LATE SATURDAY NIGHT

Well my daughter-in-law did submit the six numbers and, not surprisingly, my dream numbers did not prove precognitive. In fact I only succeeded in selecting one of the winning numbers and therefore not even a minor prize was won. See Figure 1. Nor did my other possible numbers (1, 4 and 6) make an appearance. Of course, the numbers could have been meant for Saturday April 2nd, the day before my birthday and a day that marks the 73rd solar return (when the Sun returns to the exact position that it occupied at the time of my birth).

Figure 1

Here are links to my two previous posts on the mathematics of Lotto:

• Lotto Simulations on October 20th 2018
• Lotto Loser on June 10th 2019

Of course the numbers 17 and 37 that I dreamed of may have had nothing to do with Lotto. In fact, Figure 2 gives an insight into a completely different interpretation of the numbers.

Figure 2

A shipping container is characterised by its square cross-section and so the 17 could refer to the side of this square and the 37 could refer to its length of the prism. Only two numbers are needed to define its dimensions.  When the front of the container is open and you look at it front on, the far end seems to be smaller than the open front end.  This corresponds to the twins being both 17 and yet one appearing larger than the other. 

What are the characteristics of this 17 x 17 x 37 rectangular prism? It has a volume of 10693 cubic units and a surface area of 3094 square units. If the front end is open, then the surface area is 2805 square units. How Saturday fits in with this view of things I don't know. Saturday is the sixth day of the week if we count Monday as 1, Tuesday as 2 etc. A rectangular prism does indeed have six sides.

Saturday is Saturn's day and each planet has an associated magic square. The one for Saturn is shown in Figure 3. The numbers associated with Saturn are 3, 9, 15 and 45 because the magic square is 3 x 3, there are 9 squares, the numbers in each row, column and main diagonal add to 15 and the total of all nine numbers is 45.

Figure 3

I've written about magic squares before. Here are links to these posts:

• Ultramagic Squares on November 12th 2021
• Prime Semi-Magic Squares on May 17th 2021
• Magic Squares on September 21st 2020
• Anti-Magic Squares on July 17th 2018

Other sources quote Saturn as being associated with the number 8. See Figure 4.

Figure 4: source

I seem to be going off on a tangent here so I'll stop. It has occurred to me that 17 and 37 are both reversible primes since 71 and 73 are also prime. Currently, at age 72, I'm stuck between these two primes although in less than two weeks I'll be 73. 

Let's not forget that 71 and 73 are also twin primes which fits in very nicely with the twin element in the dream. Additionally, my 73rd solar return occurs on April 2nd 2022, a Saturday. The solar return marks the return of the Sun to the same zodiacal position (12°47'07" Aries) at the time of my birth. My birthday actually occurs on April 3rd.

The shipping container may just be a symbolic representation of life on the physical plane. I am reminded of Carl Jung's description of his Near Death Experience in his autobiography "Memories, Dreams, Reflections" :

In reality, a good three weeks were still to pass before I could truly make up my mind to live again. I could not eat because all food repelled me. The view of city and mountains from my sick-bed seemed to me like a painted curtain with black holes in it, or a tattered sheet of newspaper full of photographs that meant nothing. Disappointed, I thought, "Now I must return to the 'box system' again." For it seemed to me as if behind the horizon of the cosmos a three-dimensional world had been artificially built up, in which each person sat by himself in a little box. And now I should have to convince myself all over again that this was important! Life and the whole world struck me as a prison, and it bothered me beyond measure that I should again be finding all that quite in order. I had been so glad to shed it all, and now it had come about that I along with everyone else would again be hung up in a box by a thread.

He continued:

It is impossible to convey the beauty and intensity of emotion during those visions. They were the most tremendous things I have ever experienced. And what a contrast the day was: I was tormented and on edge; everything irritated me; everything was too material, too crude and clumsy, terribly limited both spatially and spiritually. It was all an imprisonment, for reasons impossible to divine, and yet it had a kind of hypnotic power, a cogency, as if it were reality itself, for all that I had clearly perceived its emptiness. Although my belief in the world returned to returned to me, I have never since entirely freed myself of the impression that this life is a segment of existence which is enacted in a three-dimensional boxlike universe especially set up for it.

So perhaps my dream primes 17 and 37 are meant to be interpreted as the twin primes 71 and 73 with Saturday April 2nd marking my astrological coming of age 73. I realise I'm straying too far into the metaphysical here and it would be better to continue this train of thought in my blog "Mystical Meanderings".

Cubic Numbers

Today, January 11th 2016, I'm 24389 days old and what's special is that this number is 29 cubed or 29 x 29 x 29. Days like this are rare. For example, \(28^3 \) or 21592 occurred on May 10th 2009 and \(30^3 \) or 27000 will occur on March 6th 2023. Cubes of prime numbers are even rarer of course. The prime preceding 29 is 23 and \(23^3 \) or 12167 occurred on July 26th 1982. The prime following 29 is 31 and \( 31^3 \) or 29791 will occur on October 26th 2030 when I'm 81 years of age (if I make it that far). 

24389 has a surprisingly large number of entries in the Online Encyclopaedia of Integer Sequences (OEIS), 174 in fact which is unusual for a composite number of this magnitude. The first entry is for OEIS A000578: the cubes \( a(n) = n^3 \). The sequence, up to 24389 when n=29, looks like this:

0, 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389

The next entry is OEIS A030078: cubes of primes. The sequence, up to 24389, is: 8, 27, 125, 343, 1331, 2197, 4913, 6859, 12167, 24389.

From WolframAlpha, we find that 24389 is also a cube that is expressible as the sum of two squares in two different ways: 

\(24389 = 58^2+145^2  = 65^2+142^2 \)

Additionally, we find that 24389 is the hypotenuse of a primitive Pythagorean triple: 
  \(24389^2 = 15939^2+18460^2 \). So, all in all, an interesting number.