Human Genome Numbers

For want of a better term, I'll refer to the numbers that this post is about as Human Genome Numbers because the human genome is comprised of 46 chromosomes, 23 each of which are supplied by the father and mother. But first some background. I very much enjoy coming across numbers that mark my diurnal age and that have, seemingly, no interesting properties that are immediately discoverable. Such is the case today as I turn 27446 days old. My usual resources turned up nothing. 

Looking further however, I noted that the sum of this number's digits is 23, a very interesting number in its own right. I then noticed the number 46 formed by the last two digits of the number. Now 27446 factorises as follows:$$27446 = 2 \times 13723$$It can be seen that the 46 arises from the doubling of the last two digits in the factor 13723. This led me to seek out semiprimes with digit sums of 23 and with a smallest factor of 2 and a larger factor with 23 as its last two digits.

This turns out to be a rather exclusive club and there are only six such numbers in the range up to 40,000 (the top of the range that I normally work with). Here are the numbers and their factorisations:$$ \begin{align} 7646 &= 2 \times 3823 \\ 9446 &= 2 \times 4723 \\23846 &= 2 \times 11923 \\ 25646 &= 2 \times 12823 \\ 27446 &= 2 \times 13723 \\ 36446 &= 2 \times 18223 \end{align}$$So to focus on 27446 is particular, we see that:$$ \overbrace{27446}^{digits \, sum \, to \, 23} = 2  \times 13723 \text{ where 2 x 23 = 46} $$There are 82 such numbers in the range up to one million and they are:

7646, 9446, 23846, 25646, 27446, 36446, 47246, 50846, 56246, 58046, 63446, 70646, 74246, 106646, 113846, 124646, 128246, 135446, 140846, 144446, 162446, 166046, 175046, 207446, 209246, 214646, 216446, 221846, 227246, 234446, 241646, 252446, 254246, 265046, 292046, 302846, 313646, 320846, 322646, 328046, 333446, 335246, 337046, 346046, 351446, 362246, 403646, 410846, 421646, 423446, 427046, 443246, 450446, 452246, 490046, 500846, 502646, 504446, 513446, 522446, 526046, 535046, 540446, 551246, 562046, 580046, 601646, 630446, 652046, 702446, 704246, 711446, 720446, 751046, 801446, 812246, 814046, 821246, 832046, 900446, 913046, 931046

There are other ways to search for the presence of 23 and 46 in numbers but the method I've chosen identifies 27446 and that was the intent. Let's repeat the criteria for selection:

• number must have a digit sum of 23
• number must be a semiprime with 2 as the smaller factor
• larger factor must end in the digits 23 (ensuring number itself ends in 46)

Two of the previous numbers have the digit sequence 23 occurring twice in the larger factor. These numbers are:

• 47246 = 2 x 23623
• 124646 = 2 x 62323 (notice that the number itself contains two instances of 46)

One number has the digit sequence 23 and 46 in the larger factor. This number is:

• 292046 = 2 x 146023 (notice that 46 also appears in the larger factor)

Another method of selection is to choose numbers with digit sums of 23 and with 46 = 2 x 23 as a factor. This selection method yields 47 numbers in the range up to 40,000. These numbers are:

1886, 3956, 4784, 5198, 6854, 7268, 7682, 8096, 8924, 9338, 9752, 10994, 13478, 13892, 15548, 15962, 16376, 16790, 17618, 18446, 18860, 19274, 21758, 22586, 23828, 24656, 25484, 26726, 27554, 28382, 29624, 30866, 31694, 32936, 33764, 34178, 34592, 35834, 36248, 36662, 37076, 37490, 37904, 38318, 38732, 39146, 39560

Take the first member in the sequence as an example:

• 1886 = 2 x 23 x 41 = 46 x 41

The common thread that justifies the selection of what might be termed human genome numbers is that the numbers 23 and 46 must be involved in some way. This opens the door for other selection criteria but I'll leave off at this point and maybe return to this theme at a later date.

A Look at Sums of Multiples of Squares

I certainly have a deeper understanding now of what numbers can be expressed as a sum of two squares and in how many ways. See my earlier posts:

• Sums of Square Revisited
• Fundamental Theorem on Sums of Two Squares
• Numbers that are Sums of Two Squares
• Sums of Squares
• Finding Significance in Insignificant Numbers
• Sum of Two Squares
• Sum of Squares of Integers and Catalan Numbers

There's certainly some overlap in these posts but mostly they've dealt with numbers of the form \(x^2+y^2 \) rather than \(ax^2+by^2 \) where \(a \) and \(b \) are integers. It's the latter type of expression that I want to explore in this post. My interest was stimulated by today's diurnal age number, 25229, that happens to be a 4k+1 prime.

As a 4k+1 prime, it can be expressed as a sum of two squares in one way only, viz. \( 98^2+125^2 \). However, it turns out that it can be expressed a sum of two squares multiplied by varying coefficients in many different ways. I created some Sage code to investigate:

number=25229 #enter any number
x,y = var('x'),var('y')
a,b = var('a'), var('b')
for a in range(100):
    for b in range(100):
        for x in range(sqrt(number)):
            for y in range(sqrt(number)):
                if a * x^2 + b * y^2 == number:
                    if a < b:
                        print(a, b, x, y)

Running this code on the SageMathCell Server generated the following results for a <100 and b < 100. The first number represents a, the second b, the third x and the fourth y. For example, the first entry (1, 4, 125, 49) represents \(125^2+4 \times 49^2 \).

(1, 4, 125, 49)
(1, 5, 27, 70)
(1, 7, 41, 58)
(1, 13, 101, 34)
(1, 20, 27, 35)
(1, 25, 98, 25)
(1, 28, 41, 29)
(1, 29, 75, 26)
(1, 49, 125, 14)
(1, 52, 101, 17)
(1, 65, 42, 19)
(1, 85, 152, 5)
(1, 91, 127, 10)
(1, 97, 94, 13)
(2, 3, 61, 77)
(2, 11, 105, 17)
(2, 21, 110, 7)
(2, 51, 37, 21)
(3, 26, 9, 31)
(4, 25, 49, 25)
(4, 65, 21, 19)
(4, 85, 76, 5)
(4, 97, 47, 13)
(5, 6, 71, 2)
(5, 9, 70, 9)
(5, 19, 11, 36)
(5, 24, 71, 1)
(5, 46, 61, 12)
(5, 59, 45, 16)
(5, 69, 8, 19)
(5, 76, 11, 18)
(5, 81, 70, 3)
(5, 89, 69, 4)
(6, 53, 56, 11)
(7, 22, 31, 29)
(7, 29, 60, 1)
(7, 46, 53, 11)
(8, 21, 55, 7)
(9, 20, 9, 35)
(9, 29, 25, 26)
(9, 65, 14, 19)
(11, 18, 17, 35)
(11, 50, 17, 21)
(11, 98, 17, 15)
(13, 29, 36, 17) 

It would be interesting to see the full range of possibilities for a, b, x and y. However, for larger values of a and b, the SageMathCell Server seems to time out. It is easy to see how the case of (1, 4, 125, 49) comes about because: $$125^2+(2 \times 49)^2 =125^2+(2 \times 49)^2 = 125^2+ 4 \times 49^2 $$ However, in the case of (2, 3, 61, 77) where \( (\sqrt{2} \times 61)^2+ (\sqrt{3} \times 77)^2 = 25229 \), it's far from clear how the result comes about.

It's probably better to start investigating with a smaller 4k+1 prime such as 61. Now \( 61=5^2+6^2 \) and there is only one possibility with coefficients and that is \( 4 \times 2^2+5 \times 3^2 \). Similarly with \(73=3^2+8^2 \), there is only one possibility and that is \(5 \times 3^2+7 \times 2^2 \). With \(97=4^2+9^2 \), there are two possibilities: \( 4 \times 2^2 + 9 \times 3^2 \) and \(5 \times 3^2+13 \times 2^2 \).

It can be noted that 4k+3 primes such as 43, while not able to be represented as a sum of two squares, can be represented as a sum of squares with coefficients: \( 3 \times 3^2+4 \times 2^2 \). Essentially, the elements of the set of all multiples of the square numbers (let's call the set S) are being combined with themselves by addition to form a set of new numbers (let's call this set N). This set consists of \(1^2 =1\) and the multiples of 1 are the natural numbers. This of course is trivial because it means for any number n can be written as \( (n-m) \times 1^2 + m \times 1^2 \) where \( m<n \).

This result means that it is necessary to exclude 1 from our investigation and begin with \( 2^2=4 \). So all multiples of 4 are permissible, as are all multiples of 9, 25, 36, 49, 64, 81 and so on. In this scheme, the smallest possible number would be \(2^2+2 \times 2^2=12 \) with \( 2^2+2^2=8 \) being ignored since we know the rules governing the requirements for a number to be a sum of two squares. This means that coefficients must not be perfect squares and thus \(2^2+4 \times 2^2 =2^2+4^2 = 20 \) is not acceptable. Even coefficients that add to perfect squares when the squared terms are the same must be excluded.

Thus \(2^2+3 \times 2^2 = 2^2 \times (1+3) = 2^2 \times 4 = 4^2 =16 \) falls victim. From the outset, it's clear that certain numbers (apart from the numbers 1 to 11) cannot be represented by any combination of multiples of squares: 14, 15, 17, 18, 19, 20, 21 and 23. However, it seems clear that every number above 23 can be represented either as:

• a perfect square e.g. \(25=5^2 \)
• a multiple of a perfect square e.g. \( 50=2 \times 25 \)
• a sum of two different squares e.g. \(25=4^2+3^2 \)
• a sum of the multiples of two squares e.g. \(25=4 \times 2^2 +3^2 \). 

I've not proved this by any means but it seems highly likely, given the increase in possibilities as the numbers get larger. So the main takeaway from all this is that \( ax^2+by^2=23 \) is the last ellipse in the family of ellipses \( ax^2+by^2=N \), where N is a natural number and a>1 and b>1, that does not have integer solutions for a, b, x and y.

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.