# Junction Combination Theorem

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Suppose that, $a_{k_1} + s_d(a_{k_1}) = a_{k_2} + s_d(a_{k_2}) = \dots = a_{k_r} + s_d(a_{k_r}) = N$ for $k_1, k_2, \ldots k_r > 1$.$s_d(n)$ denotes the sum of the digits of n. Thus all of $a_{k_1}, a_{k_2}, \ldots , a_{k_r}$ yield $N$. That is, $N$ has $r$ “generators”. Numbers with more than one generator (i.e. $r>1$) are called “Junction Numbers” by Kaprekar.

Let $N-1$ and $N+1$ be two “Junction Numbers”, such that they have equal number of digits, say, $d$ digits. If $P_1, P_2, \ldots$ be the generators of $N-1$ and $R_1, R_2, \ldots$ be the generators of $N+1$.
Then,

$1(0)_{(1)_d} P_1 \quad, \quad 1(0)_{(1)_d} P_2 \quad, \quad \ldots$

$(9)_{(1)_d} R_1 \quad, \quad (9)_{(1)_d} R_2 \quad, \quad \ldots$

give the generators of $1(0)_{(1)_d} N$.

Let us consider an example to understand this theorem.

Consider two Junction Numbers 519 (generated by both 498 and 507) and 521 (generated by both 499 and 508).

Here $N-1 = 519$ and $N+1 = 521$

$\therefore P_1 = 498 \quad, \quad P_2 = 507$
$\therefore R_1 =499 \quad, \quad R_2 = 508$

$d = 3 =$ Number of digits in $N-1$ or $N+1$ or $N$.

$\therefore (1)_d = (1)_3 = 111$
Then according to the theorem,
$1(0)_{111} 498 \quad, \quad 1(0)_{111} 507$

$(9)_{111} 499 \quad, \quad (9)_{111} 508$

give the four generators of $1(0)_{111} 520$.

$1(0)_{111} 520$ is a number with 115 digits

Kaprekar named this theorem as Kaprekar’s Last Theorem in 1962, when he was seriously ill and feared that his death was nearing. He miraculously recovered and named it as Junction Combination Theorem.

Smallest “Junction Numbers”

• Smallest “Junction Number” with two generators is $\mathbf{101}$
• Smallest “Junction Number” with three generators is $\mathbf{10000000000001}$ or $\mathbf{1(0)_{12}1}$
• Smallest “Junction Number” with four generators is $\mathbf{1000000000000000000000102}$ or $\mathbf{1(0)_{21}102}$

# Happy Birthday Kaprekar

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Today is 110th birthday of great Indian mathematician D. R. Kaprekar

Dattaraya Ramchandra Kaprekar (17 January 1905 – 1986) was an Indian recreational mathematician who described several classes of natural numbers. For his entire career (1930–1962) he was a schoolteacher at Nasik in Maharashtra. He published extensively, writing about such topics as recurring decimals, magic squares, and integers with special properties. Kaprekar was once laughed at by most contemporary Indian mathmaticians for his so-called ‘trivial’ play with numbers. It required G. H. Hardy to recognize Ramanujan while Kaprekar’s recognition came through Martin Gardner (he wrote about Kaprekar in his “Mathematical Games” column in March 1975 issue of “Scientific American”)

Here I would discuss a mathemagical trick re-disvovered by Kaprekar called “Gap Filling Process” (though claimed to be present in vedic mathematics)

Gap filling process

This process is magical one and will make you Mathemagician

Let, $(a)_n$ stand for $a$ repeated $n$ times (called Repunit $a$).

Then, we shall denote $(a)_n ^m$ for $m$-th power of $(a)_n$

$(9)_n ^m$ can be obtained by remembering the expansion for $(9)^m$ and inserting in the gaps between digits of expansion of $(9)^m$ with the numbers $(9)_{n-1}$ and $(0)_{n-1}$ alternately, beginning from left to right. No gap is counted after the unit digit.

Let’s see an example:
Find the value of $(99999)\times (99999)\times (99999) = (99999)^3 = (9)_5 ^3$.

Even my scientific calculator fails to calculate this exact value !

We know $9^3 = 729$

Then the gaps are: $----7----2----9$
Now fill the blanks alternately with $(9)_{5-1} = (9)_4$ and $(0)_{5-1} = (0)_4$.
We get: $9999\textbf{7}0000\textbf{2}9999\textbf{9}$
Hence, $(99999)\times (99999)\times (99999) = 999970000299999$

# You can think Mathematically

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This is an extension of my last blog post.

My last post was my perception and quest for obtaining answer for a difficult question.

But I do asked motive of living to a few other people, the answers (different from mine) were:

• To earn money, spend it on luxuries (by imitating them to be necessities) and to find a girl/boy [not necessarily a life partner]
• To do serious research, earn fame and money.
• To enjoy their work (no fame/money/family)

I am no one to conclude that which reason is best for being alive, but would like to suggest everybody to think atleast once “Why I am alive?” and try to come up with some answer, as I believe that living without any reason is as good as being dead.

You may also wonder why I didn’t include “money” in my motive for living, this is simply because I believe that:

The richest person is not the one who has the most, but the one who needs the least.

But most of times people try to escape from this question by indulging in various types of addictions. In college people have various types of addictions like:

• Addiction to high CGPA and building CV

I am addicted to writing senseless long e-mails, but now working to get rid of it, by speaking more (than writing)!

Never be depressed, always remember

To do nothing no effort is needed but to do something a lot of effort is needed.

If you don’t believe me simply watch this video on struggle and fame of Albert Einstein:

I would like to end this blog post with my favorite quotation:

A Mathematician is dead the day he/she stops doing mathematics.
Paul Erdős

# Inquisitive Mathematical Thinking

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Being a budding mathematician, I took these words seriously:

Mathematics is not only solving for x but also figuring out y (why).
Arthur Benjamin

To ask ‘Why?” is easiest and most of times a difficult question to answer. Also mathematics encourages asking “Why?” after each argument.
For example:

• Why earth rotates in west to east direction?
• Why people don’t like me?
• Why force is equal to mass times acceleration?

But not all questions can be answered, as there are (proved to be) unanswerable questions  in mathematics like Continuum Hypothesis.

So one fine morning I got up and asked myself “Why I am alive?” .  There must be some reason behind this as I am alive for last 18 years. [This is just an innocent question which doesn’t implies that I want to die!]

I have tried reading some holy texts and they say that:

Meaningful life is living for others.

So in that case four possible answers came to my mind:

First possible answer came to my mind was “I am alive for my mother“. So to confirm it I called her (from my mobile phone) and realized that it is not a permanent answer (what if she dies before me?).

Second possible answer came to my mind was “I am alive for contributing something to mathematics“. Now mathematics is immortal, hooray! But at same time takes me away from real world (as I am interested in pure mathematics rather than applied mathematics). Which make me feel lonely and I have an urge to take drugs [like Paul Erdős]. Actually there is always a big threat behind worshiping the work too much. So I was not very much satisfied by this answer also.

Third possible answer can to my mind was “I am alive to show others the beauty of nature“. Let me illustrate this :
Whenever NISER students go to canteen they encounter a large number of Mimosa pudica plants, but I wonder how many of them notice such wonderful plants surrounding them.
But idea of beauty being abstract is different for different people and should be discovered by one on his/her own. I can try to motivate others and keep myself satisfied that I am contributing something to mankind. So far I was very convinced by this answer. But I still thought further.

Fourth possible answer came to my mind was “I am alive to find and interact with my best friend“. Now this search can itself be lifelong, challenging and luck based. Also then I would depend on others for happiness. Thus I won’t be independent. I will be more emotional. I think this is definition of human. But still he will be mortal, and cause similar kind of problem as in case of living for my mother.

Fifth possible answer came to my mind was “In am alive to find and interact with a life long partner“. Now this relation is some what immortal, at least as compared to my lifetime. But this search is in itself stressful and purely luck based and may need a divine intervention. Here again I would depend on someone else for happiness. This is most dangerous option , I can ever think of (the romantic songs are just spoilers). As a wrong decision can be fatal. Let me illustrate this with a real life story:
Évariste Galois apparently fell in love with Stephanie-Felice du Motel, the daughter of the resident physician. Galois exchanged letters with Stephanie, and it is clear that she tried to distance herself from the affair. The name Stephanie appears several times as a marginal note in one of Galois’ manuscripts. Galois fought a duel with Perscheux d’Herbinville , the reason for the duel not being clear but certainly linked with Stephanie. Galois was wounded in the duel and was abandoned by d’Herbinville and his own seconds and found by a peasant. He died in Cochin hospital on 31 May and his funeral was held on 2 June.
After thinking for a while I could rank my preferences for being alive:

1. To show others beauty of nature
2. To contribute something to mathematics
3. For my mother
4. To find  and interact to a life long buddy
5. To find and interact a life long partner (being a ‘human’ still couldn’t rule out this option!)

I must accept that life is beautiful just because it is dynamic. So whatever reason I may state for being alive is going to be difficult. The more chaotic my life is, more blessed I am.

# Real vs. Abstract

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I am a budding mathematician and feel the conflict of abstract and real world on daily basis. So I have decided to clarify this.

The word ‘mathematics’ comes from the Greek mathemata , meaning “things that are learned”. Greeks included study of numbers, space, astronomy and music as mathematical subjects.

The word ‘science’ comes from Latin scientia, meaning “knowledge”.

The word ‘art’ comes from  Old French art, meaning “skill as a result of learning or practice”.

But mathematics converts the world of reality into a world of concepts and studies the laws of abstract concepts which governs the real world. So I believe that mathematics is lesser of a science and more of an art . For me mathematics is an abstract bridge between science and art.

Bertrand Russell said – “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”
As mathematics tells us nothing about the real world , but states that one thing follows from another in a certain way.

R. M. Bhagwat had written in his book “Everyday Mathematics” – “Mathematics can be said to be a mirror in which life is reflected. A mathematician facing a real problem can then, like Alice in Wonderland, wander in the mathematical image of life, allow his imagination to work, solve the mirror problem and then return back to real life with the real answer.

So, I have come to a conclusion that a mathematician is blessed to have relationships in both abstract and real world. In pure mathematics we spend more time in abstract world and in applied mathematics we spend more time in real world.

I believe my life as a mathematician would be “Experimenting – Conjecturing – Proving”.

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You may think that I discuss more of philosophy than mathematics on my blog, but please be patient as a pure mathematics article is under development and will be published by end of this month. Till then I have these fillers!

# New Year Resolution

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I have been in Bhubaneswar for past six month and believe that i will have a very long stay here, thus its necessary to learn one of the languages spoken in Odisha called Odiya.

So, this year I plan to start the process of  removal of language barriers, at least in study of mathematics.

I believe that by the end of this year I will be able to decode these three books.

Happy New Year to all!