Looking back into the past, I have some good moments which didn’t seem good at that moment, but now I got to say that they are the best things that happened to me when compared to my current situation. I had been doing really good in my past but as a result of some wrong decisions, I think I already had faced the worst phase of my life 2 years back. I am not sure about the worst part though in my previous sentence. These wrong decisions are still haunting me and I think they will haunt me for the rest of my lifetime.

I feel like losing in my life until I grabbed a job opportunity in one of the good companies. I considered this as my first win in the recent time and I started fighting real hard to get to a good position. At one time, this looked like a gettable thing within a short span of time. Things turned around a little while back after the so called appraisal process. I feel depressed a lot when there is difference between what I expected and what is real. The thing that hurt my ego the most is the feeling that I am not worthy enough than the people who are less experienced than me and I am working better than them.

All these thoughts flood my mind whenever I am free and they depress and demotivate me. I am feeling that I am on the edge of loosing all hope what I have left with, as usual. I may have to be more optimistic now and not compare myself with others and start concentrating on the career although this is difficult to implement. After all, I think its time to start a fight with myself before resuming the fight with others and gain control over self which I lack at the moment and also gain self confidence that I can achieve greater things and strive for dedication and hard work to complete the tasks that are started and finally concentration which eases completion of tasks. This is going to be a big fight for sure and the one which I already started fighting long ago without me knowing it and it will be the best moment of my life if I win this fight.

WISH ME LUCK!!!!

But I won’t say that all the songs are catchy the first time you listen to them. I liked piliche and makathika 3rd or 4th time I listened to them.

Here is the review of individual songs.

Sada Siva: I would say this is the best song of the album. Everything is perfect. Rating: 4.5/5

Bhoom Shakanaka: A routine song by mani and an average song. Rating: 3/5

Piliche: Melody song and it becomes very slow at some places. Also looks similar to “Nammavemo” from Parugu Rating: 3.5/5

Makathika: Good and routine song by mani. Rating: 3.5/5

Sunday Monday: Below average song and least likable song of the album. Rating: 2.5/5

Taxi Taxi: Think its hero intro song and it has good beats and also looks different. Rating: 4/5

Overall Rating: 3.5/5

I read that “Mahesh Khaleja” title is also objected by the Vijaya Bhaskar Reddy and it seems he went to the court. I hope that Khaleja will be released according to the date and will be a hit inspite of the competition from Brindhavanam and Robo.

d(x): number of positive divisors of A

If x and y are positive integers and co-prime to each other, then

d(x)*d(y) + 1 = Σ P(z, x){z: z>0 and z|y} + ΣP(w, y) {w:w>0 and w|x} <—- Result

Proof:

Identity(1):
If x, A > 0 and x is a divisor of A, then P(x, A)+P(A/x, A) = d(A) + 1

Identity(2):
If x, A > 0 and x is not a divisor of A, then P(x, A)+P(A/x, A) = d(A)

P(x, x*y) = d(x) + number of divisors of x*y but not divisors of x and less than x

Consider a divisor z (not 1) of y. All the multiples of z less than x are divisors of x*y but not x.
Number of multiples of z less than x
= Number of divisors of x less than x/z
= Number of divisors of x not greater than x/z
= P(x/z, x)

As y is co-prime to x
P(x, x*y) = d(x) + Σ P(x/z, x) {z: z > 1 and z|y}
P(x, x*y) = Σ P(x/z, x) {z: z>0 and z|y}

From Identities (1) and (2)
P(x, x*y) = 1 + Σ (d(x)-P(z, x)) {z: z>0 and z|y}
P(x, x*y) = d(x)*d(y) +1 – Σ P(z, x) {z: z>0 and z|y}

Similarly
P(y, x*y) = d(x)*d(y) + 1 – Σ P(w, y) {w: w>0 and w|x}

Add these both equations
P(x, x*y) + P(y, x*y) = 2*d(x)*d(y) + 2 – Σ P(z, x) {z: z>0 and z|y} – Σ P(w, y) {w: w>0 and w|x}

From Identity (1)
d(x*y) + 1 = 2*d(x)*d(y) + 2 – Σ P(z, x) {z: z>0 and z|y} – Σ P(w, y) {w: w>0 and w|x}
If x and y are co-prime to each other, then d(x*y) = d(x)*d(y)
d(x)*d(y)+1 = 2*d(x)*d(y)+2- Σ P(z, x) {z: z>0 and z|y} – Σ P(w, y) {w: w>0 and w|x}

Σ P(z, x) {z: z>0 and z|y} + Σ P(w, y) {w: w>0 and w|x} = d(x)*d(y) + 1

Difficulty: Medium

There is a square field of length ‘a’ (any positive real number) whose corners are A,B,C,D in the clockwise direction. Two persons started running from corners A and C along the circumference of the square field in the clockwise direction with integral speeds of v₁ and v₂.

Initially, at time t=0, distance between them is AC=√2a. Find the time t in terms of a, v₁ and v₂ when the distance between them is √2a for the next time.

Comments/Solutions are welcome.

Difficulty: Medium

There are 60 records given to you which correspond to a prisoner who is imprisoned for 60 days. He has 6 relatives of which one of them visits him daily and the others visit him every i^th day from the day of his imprisonment (i=2,3,4,5,6 for these 5 relatives).

Every record is sealed with the day number on it which implies the number of days he is jailed when the record is filed with the names of visitors on that particular day. You have to make a new record which should be filled with the following details:

visitor name – number of visits after 1^st 60 days

Assume that no other relatives visited him at all, names of these 6 relatives are different and you don’t know their names. Find the minimum number of records that need to be checked to make the new record correctly. Find the number of ways you can choose minimum number of records and still you can make the new record correctly.

Comments/Solutions are welcome.

Difficulty: Medium

There are two rooms A and B which are not connected and whose entrances are closed. Each of the rooms has 5 bulbs inside the room and 10 switches outside the room corresponding to bulbs of both the rooms.

The entrances will stay opened if and only if exactly one of the switches outside A and one of the switches outside B are ON. If both the switches that are turned ON correspond to the same bulb then the bulb will be OFF.

You are asked to find the bulbs corresponding to each of the switches outside rooms A and B. Using optimal methods, find the minimum and maximum number of trials required to complete the task.

Note: A trial is defined as turning one of the switches outside A and one of the switches outside B ON and checking the bulbs in the rooms A and B. There is no way to find which bulbs are ON from outside the rooms. The bulbs are at a reasonable height so that you can’t touch any of them.

Bonus:Assume that you are choosing both the switches(one from A and one from B) randomly i.e., not using optimal methods, find the probability that the task is completed in minimum number of trials.

Comments/Solutions are Welcome.

PS: I will be giving one puzzle everyday from today. Enjoy solving the puzzles.

Recently, I submitted a paper to American Math Monthly about a theorem on Multiplicative Order. I think its new and it leads to Multiple-key Cryptography, but it was rejected and the mail, that was sent to me is as follows:

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Dear Mr. Yalavarthi,

I am writing about your paper “A Theorem Related to Multiplicative
Order” (manuscript #07-0425), which you submitted to the American
Mathematical Monthly. I have read your paper and discussed it with
members of the editorial board.

Articles in the Monthly must meet high standards of exposition. The
English must be fluent and enjoyable to read, and the mathematics must
not only be correct, but also be presented in a way that is clear and
inviting to a wide range of readers. We don’t believe that your paper
meets these high expository standards, and therefore we will be unable
to accept it for publication.

These are difficult decisions. The Monthly receives a large number of
submissions each year, and we are able to publish only a small fraction
of them. Page limitations force us to turn away many fine papers.

Thanks for thinking of the Monthly.

Sincerely,

Daniel J. Velleman
Editor

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Of course, I know that I am not fluent in English. Now, what should I do with this paper?

Throw it in thrash? Suggestions plz. You can tell whether the paper is useful or not and the paper is attached.

After having finished reading about 20 pages, I felt that I need to refresh before beginning another session. It was around 7 and I am feeling hungry, so I went out with my friends to eat. My bad luck started here. I ordered different items from those of my friends’. So, I knew that my turn was last, but it was still worse than that. My order had been served to another and I was starring at that guy, speechless.

After that, I was still off the mood. Then I thought I might relieve a bit more by watching a movie or just some part of it. There came these awesome threesome movies. I thought they were lame after watching them.They are as follows:

1) Imagine Me & You: I was impressed by the scene when heroin wore her wedding dress as she looked pretty well in that. There is a girl who supplies flowers at her wedding party. When the bride is reaching the groom, then she and this flower-girl exchanges looks with each other and she (flower-girl) is going outside the church. I thought there will be some twist in the story. Later I got to know these facts, they just met there in the church. When heroin invites other to her house for dinner. In her talk with heroin’s husband, this flower-girl says that she is lesb. This sounded aweful to me. I dragged the bar till the end to find out that they became one and lived happily ever after. This s..ks.

2)Friday the 13th ***: I dont exactly remember the title. I expected it to be a horror movie, but it wasn’t. The deaths were horrible in this way such they just shout before they are killed not even by a horrible looking creature but by an old woman.

3) The last but not the least is “One Last Thing”. It was somehow better than the other two. There is a guy of around 15 years of age who knows that he is gonna die in few days. As his last wish he wants to spend sometime(donno exactly) with super model who is married and divorced too. Finally his last wish comes true and he dies after that.

My mind is completely blocked after watching these movies. Right after that I went straight to my room and slept. Somehow I managed to attempt 70 marks out of 90 in DIP, but I dont think I will get more than 50 in that exam. Hoping that my luck decides my score in that exam, I am ending this post.

1)For our BTP(B.Tech Project), I have to find fast and efficient ways to find modular inverse. Then I came out with this interesting result.

(p-1)!modp=-1(mod p)

This happened to be the "Wilson's Theorem"

2)Next I studied about Fermat's Little Thoerem and its use in modular exponentiation method and there I came across Totient function which gives no. of relatives to x and less than x. There I found out another interesting result.

"Sum of totient values of all the factors of a number is the number itself". 

I found this result in mathworld and got disappointed. This result has no proof there.
Finally I learnt a lesson from my experiences. The lesson is that number theory had been explored to its maximum depth. I have 2 proofs for the second one.I think atleast one 1 proof of them is new. So, I have to try writing a research paper for that identity.

You will find number theory interesting when you are new to it. You will find many intereting results when you are getting deeper into the subject, but none of them will be new.

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