Monday, April 29, 2019

Babies, P-A-C And Communism

There are two ways of looking at our life. One way is to look at whether something is comfortable to one, whether it disturbs one's peace of mind and whether one needs it. This doesn't keep an accounting of what one person did to another, whether one owes something back etc. This is the way babies look at life. They do not understand that mom is busy. If a baby needs something, the baby howls expecting immediate attention.

The barter mechanism, on the other hand, looks at exchange of goods and services as the mechanism to address deals between people. This doesn't consider one's needs or pains as the primary factor to deal with people. 

"I have constraints, so i cannot help you. I have needs, so you have to help me" is the way babies deal with the world. Not just babies. Any adult too, when overwhelmed by pain or desire, acts exactly the same way. This is nothing but communism in action.

Focus on pain and desire forms the basis of communism (and Venusian traits). This is also the baby or Child state.

Focus on responsibility forms the saturnine trait, which is the Parent state.

Focus on Laissez-Faire is being in the Adult state. Playing Games as per Game Theory seems to be a combination of Child-Adult started strictly excluding Parent state.

Additional Reading
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Venus, The Jack Of Line Function

Depth in a field and passion in it are incompatible with a zest for life. The zest or josh comes from Venus and this goes better with program management of multiple functions than depth in any one. 

Venus is the Jack that manages All Trades and heads the line function. Staff function is created by centralization. It is associated with depth in one field and is the antithesis of Venus.
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Sunday, April 28, 2019

Stress Induced Mutation

After writing about need (Venus) based evolution that I suspected earlier here, I read this:
https://www.wired.com/2014/01/evolution-evolves-under-pressure/ "Cells actually decide to turn up their mutation rate when they are poorly adapted to the environment. "

 X ray on bacteria in the 1950 created consistent replicable mutations. I believe that stress and need cause mutation. Hence animals high in the future chain don't (need to) mutate much, example being crocodiles, maybe lions?
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Tuesday, April 23, 2019

Kunti And Karna

In Mahabharatha, the story of the relationship between Kunthi and her illegitimate son Karna is interesting. 

After he gets to know that Kunti is his mother they have a affectionate reunion. Soon Kunti asks Karan to leave the Kauravas and join her other 5 sons, the Pandavas. He explains why he is indebted to Duryodhan, the eldest of the Kaurava brothers.

His mother asks him two favors. 1. That he should not fight against or kill any Pandava other than Arjuna who is Karna's sworn enemy. 2. That Karna should not use the deadly Nagastra (a guided missile!) more than once against Arjuna.

Both these favors together more or less eliminate the possibility of Karna doing much harm to the Pandavas while the favors  he granted endanger Karna's own life. It's noteworthy that Kunti didn't grant Karna nor did Karna ask his "new mother" any favor to help protect his own life.

Of course Karna, being the person he was, unhesitatingly grants his mother the favors she asks of him. And in the war, he is killed by Arjuna. 

What I find interesting is that Kunti ensured that her 5 legitimate sons were protected , even if it meant the end of her illegitimate and eldest son. While she makes a show of all sons being equal in her eyes (that the new bride was to be shared by her 5 sons, read here) she makes a clean distinction between the eldest and the younger 5 sons. What is remarkable is her calculating mind at work when she asks Karna the two favors while at the same time having an emotional reunion with him. 
This story is as per the Tamil film Karnan. Quote from the movie's wiki "Karnan is based on the life of the character Karna from the Hindu epic Mahabharata. B. R. Panthulu, who directed and produced the film under the banner Padmini Pictures, had collected most of his information from scholars Kripananda Variar and Anantarama Dikshitar.

I am reminded of mother birds and mother crocodiles that sacrifice their own young runt in order to survive or to increase the survival probability of her stronger children.

It's sad and thought provoking that Kunti saw a runt in Karna.

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Sunday, April 21, 2019

Dallying With Sacrifice

A friend of mine F,  commented to me that her sister S recently told her that she, S, had been in a relationship with a man M for about 30 years. The sister and and the man never got married, they live in different cities. And that M was from a different community which my friend's parents would not have liked at all.

My friend told me apologetically that she had often taken up her sister's time with some trifles not knowing that S belonged to someone else, that S had a significant other in her life all these decades.

I asked my friend whether her sister was a person who gave two hoots what their parents felt. My friend thought for a second and replied no. So the question of sister "sacrificing" for the sake of parents was obviously not the case.

I told my friend that there is likely a reason for S and M not to have gotten married for so long. And that is that they probably wanted nothing more than a good relationship - maybe something like FWB. Maybe neither wanted anything more from each other. 

And I added that my friend taking up her sister's time, though unknowingly, was perhaps a welcome diversion in her sister's life. The sister could probably tell the man that she was busy helping my friend. Maybe the man had a similar thing in his life. My friend and her parents provided a legitimate diversion to the sister. Maybe a much needed diversion. 

 If someone is unable to find time for a particular person or activity (POA) then that POA was never that important to them. If that person actively finds other activities to do while keeping the POA under wraps then that POA was more like a drug - nice to dally with but not nice to have full time.

The idea of people sacrificing their life for the sake of someone other than their children is remote. 

My friend felt that her sister used to help her, F, out at some cost to her own relationship. Na. No way. It was likely convenient to S. That  definitely doesn't reduce the value of the help provided.

That leads to an aphorism. If someone seems to be sacrificing their life, they aren't. They are doing it for some personally convenient or desirable reason. 

I finally asked my friend if she would feel better if her sister had (1) sacrificed her own life, not getting married nor having her own family, for the sake of parents or (2) if sister had a personal reason for not getting married that had nothing to do with the parents.

My friend replied that the second option was better. She had also earlier told me that her sister was also unlikely to have resorted to option 1.

This being the case I wonder how my friend, an intelligent woman, thought that the sister, at great cost to herself, was handling multiple fronts without letting on that she was in a relationship.

Women love the abstract idea of a sacrifice but at a tangible level they hate it. I have often seen my friend and other women indulge in such emotional orgasms - hoopla about sacrifices. And often times when I lead them through the circumstances and explain the rationale behind said "sacrifice", they realize half heartedly that there never was a sacrifice. 

 I say half heartedly because they, people like my friend, would repeat the "sacrifice" incident again to another gullible audience eliciting orgasmic ooohs and aahs.

 Reality is fairly depressing. No wonder the adventurous take to poetry. And we resort to opinions from our S1. 

Additional Reading
https://vbala99.blogspot.com/2018/02/vikram-or-vetaal.html - More on sacrifices
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Saturday, April 20, 2019

Mulling Over Mueller

This post was written after Barr submitted a brief summary of Mueller report. 

 After the Mueller report is published, it's like recovering the CVR after a plane crash - a whole lot of data and a lot of opinions from every side.

But what exactly do these opinions mean? Which one is right? Are they all right?

WP and NYT as usual are speaking against Trump and Barr. WSJ is speaking for Barr.

Here is a sample of articles published after the report.


  • What is the big deal about his tax report? It's likely he would have fudged and cheated? What does it prove? How does it help?
https://www.washingtonpost.com/opinions/2019/05/23/wall-concealment-trump-built-around-his-finances-is-beginning-crumble/


https://wapo.st/2QB3jgV
The (US) laws that exist today only show that they are inadequate to apply to a character like Mr Trump. An important question in this regard is: What would the Republicans had felt about Mr Trump if he had been a non-Republican? And what would Democrats have felt if he had been a non-Democrat?

 If Trump can be manipulated by Russia to do things that are adverse to USA, is he appropriate for his position?
If social media, in its current Avatar, is detrimental to US interests, is it appropriate to have?
If drugs are adverse to a country's interests should they be allowed?

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Tuesday, April 9, 2019

Prime Numbers And Gap and R1 - Ready Reckoner

This post has to be read after the previous post on the same topic. https://vbala99.blogspot.com/2019/04/estimating-next-prime-number-conts.html


Having understood Gap and R1, we proceed further. Take the prime 523 which has a high Gap. Note that there are other primes with even higher Gaps. And the only prime N1 whose Gap is higher (though marginally) than square root of N1 is 113 as we saw in the previous post.


523            R1+n*factor (only evens upto 38)
R1 
3: 2              8 14 20 26
5: 2            12 22 32
7: 2            16 30
11: 5          16 38
13: 10        36
17: 4          38
19: 9          28
23: 6

Given below is a ready reckoner for prime factors 3,5,7,11,13 and sample data for prime factors 17,19,23. 
It lists the even values of R1+n*factor for each combination of factor and R1. Obviously R1 can never exceed the factor.
Let's assume there is a prime number N1 for which we have found out the R1 values for each prime factor.
Take the first row below. When R1 for factor 3 is 1, the even numbers in R1+n*factor are 4, 10, 16, 22, 28, 34, 40 etc upto the square root of N1
Take the fourth row. When R1 for factor 5 is 2, the even numbers are 12, 22, 32 etc.



3: 1             4 10 16 22 28 34 40...
3: 2             2   8 14 20 26 32 38...
5: 1             6 16 26 36...
5: 2             2 12 22 32...
5: 3             8 18 28...
5: 4           14 24 34...
7: 1             8 22 36...
7: 2           16 30...
7: 3           10 24 38...
7: 4           18 32...
7: 5           12 26...
7: 6           20 34...
11: 1         12 34...
11: 2         24...
11: 3         14 36...
11: 4         26...
11: 5         16 38...
11: 6         28...
11: 7         18 40...
11: 8         30...
11: 9         20...
11: 10       32...
13: 1         14...
13: 2         28...
13: 3         16...
13: 4         30...
13: 5         18...
13: 6         32...
13: 7         20...
13: 8         34...
13: 9         22...
13: 10       36...
13: 11       24...
13: 12       38...
17: 1         18...
17: 2         36...
17: 3         20...
17: 4         38...
19: 1         20...
19: 2         40...
19: 3         22...
19: 4         
19: 5         24...
19: 6         
19: 7         26...
19: 8         
19: 9         28...
23: 1         24...
23: 2
23: 3         26...
23: 4         
23: 5         28...
23: 6         
...
This lookup is provided in the LOOKUP sheet in https://docs.google.com/spreadsheets/d/1nppxKrfkNOpI22NfzWP_ZNerC9wzvIBSJm8WOb9BqMQ/edit?usp=drivesdk. The Prime Factor sheet calculates the next prime when given a prime.

When you have a prime that has an R1 of 1 for factor 3 and R1 2 for factor 5 then take the union of the sets in the first and fourth rows and find the least even number. Of, course you have to consider ALL primes higher than 3,5 also upto the square root of the current prime N1 that you are considering. If N1 is 523, you have to consider prime factors upto 23. For each prime factor, you will take one row depending upon the value of R1. If Oppermann's Conjecture is true, then you will find an even number, less than the square root of N1, that doesn't belong to the union of the sets as explained before. The reverse is also true. If for every N1, you are always able to find such an even, then the  Conjecture is true. Incidentally this even number is nothing but the Gap.

The only trivial exception as I mentioned earlier is N1= 113 for which the Gap (=14) is slightly higher than the square root of N1.

Additional Reading
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Saturday, April 6, 2019

How To Evaluate Remainder And Quotient For Another Number (Devisor)

If you already know the quotient and reminder when you divide x by a, then what will be the quotient and reminder if you divide x by b?

All numbers discussed here are integers.
Suppose there are there numbers x, a, b
When you divide x by a, the reminder is a1 and quotient is a2.

If you now divide x by b, what will be the values of the new quotient b2 and reminder b1 in terms of x, a, b, a1, a2?
What will be the new reminder if you divide y by a is a far simpler problem explained in 
https://vbala99.blogspot.com/2019/04/estimating-next-prime-number-conts.html. Though the associated discussion on prime numbers is more involved.


Coming back to our problem, assume that the values of x, a, b, and hence of a1, a2 are known.

a2 = (x-a1)/a
x + (b-a)*a2 when divided by b will give the same reminder a1 and same quotient a2.
For example assume x=13, a=3, b=5.
a1=1, a2=4

 Let's term (b-a)*a2 as c

x + c = x + (b-a)*a2 = 13+(5-3)*4 = 13+8=21.
If 21 is divided by b=5, the quotient is 4 and reminder is 1. As before.

Remember values of a1, a2. We have to determine the values of b1, b2 when x=13 is divided by b=5.

Now let's focus on the number that was added to x: c = (b-a)*a2 = 8

Let's divide c by b=5
The quotient is 1 and reminder 3.

When x was divided by a=3, the quotient was 4 and reminder 1.

To determine the b1, b2 we take the first set of quotient and reminder {4,1} and reduce the second set {1,3} from it.

We get the new set 4-1=3 and 1-3=-2
The new quotient and reminder are 3, -2.

3*5 +(-2) is indeed 13. Rather, when 13 is divided by 5, b1=3 and b2=-2.
But then reminder should be positive.
So we subtract 1 from b1 and add b to b2.

Final answer b1=3-1=2 and b2=-2+5=3.

 If we put it in algebra:
If the set of quotient, reminder when x is divided by a is {a2, a1} and the set when c = (b-a)*a2 is divided by b is {c2,c1}

 Then when x is divided by b the quotient and reminder are {a2-c2, a1-c1} ...... Eqn (1)

 In case a1-c1 is negative then reduce a2-c2 by 1 and add b to a1-c1. ..... Eqn (2)

 Let's apply this to an example.
x=167, a=8, b= 17
a2=20, a1=7 (because 20*8+7=167)

 Now,
c = (b-a)*a2 = (17-8)*20=180
When 180 is divided by 17, the quotient and reminder are {10,10}

 So the final set, applying Eqn (1) is {20-10, 7-10} = {10, -3}. Note 10*17+ (-3) = 167 = x

 Since -3 is less than 0, we apply Eqn (2) and get the corrected quotient and reminder set:
 {10, -3} = {10-1, -3+17} = {9, 14}

 Check: 9*17 + 14 = 167

 Simple.
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Thursday, April 4, 2019

Estimating The Next Prime Number... Rewritten

The first version of this post was written in 2016. I recently rewrote to make it more comprehensive.

 As far as I know, there are a number of algorithms to determine whether a number is prime. This post is about determining the Gap that directly leads to the next prime number when a prime number and all smaller prime numbers are known. The procedure is deterministic. The next number found using this procedure will definitely be a prime. Matter of fact, this method works even if you start with any odd number, not necessarily a prime. The only requirement is that you need to have all pedigree info, i.e., all smaller prime numbers - right upto 3. 

 This post determine Gap (to find the next prime) for a given known prime. This is done by evaluating something called R1 for the current prime number. The difference between two successive prime numbers is called Gap. As per Oppermann's Conjecture. the gap size would be on the order of
    g_n < \sqrt{p_n}\, . This post is about estimating the Gap and not about the Conjecture per se.
    Read also the Table "80 known maximal prime gaps"  https://en.wikipedia.org/wiki/Prime_gap. We will come back to Gap later in this post.


     Let's start with the prime number 67

    Our objective is to determine the next prime number without going through the division route.

    Though we should use prime factors, I have used odd numbers as factors. This is a good starting point, though not as efficient.

    Factors to be considered for 67 are 3,5,7,9,11 (We have taken 11 also though it could have been excluded.)

    We now define R1.
    R1 is another form of reminder when x is divided by y (both integers. All numbers are integers only unless otherwise specified). 

    R1 is the minimum number to be added to x to make the sum a multiple of y.
    R1 values for 67 when divided by 3,5,7,9,11 are respectively: 2, 3, 3, 5, 10
    Example:
    67 +10=77 which is a multiple of  11
    67+3=70 which is a multiple of both 5 and 7. 

    We now start from 67:
    Find the first even number 2,4,6,8 etc which does NOT match any of the R1 (=2,3,3,5,10) AND SHOULD NOT MATCH ANY OF THE SUM OF FACTOR AND IT'S R1. The set of Sum of R1 and it's factor are {3+2, 5+3, 7+3, 9+5, 11+ 10}.
    In this case it is 4. Add this 4 to the current prime. 
    67+4=71. 

    71 hence is the next prime. BECAUSE THE SMALLEST EVEN NUMBER 4 IS NOT IN THE SET OF R1 NOR IN THE SET OF FACTOR + R1.
    AND HENCE WHEN ADDED TO 67 WILL NOT BE EXACTLY DIVISIBLE BY ANY OF THE FACTORS.

    Now start with 71 and repeat.
    For 71, R1 = 1,4,6,1,6. THE SET OF FACTORS + R1 IS {3+1,5+4,7+6,9+1,11+6}.
    The smallest even number not in this set of R1 is 2.
    Hence the next prime is 71+2=73.

    Now start with 73
    R1=2,2,4,8,4. THE SET OF FACTORS + R1 IS {3+2,5+2,7+5,9+8,11+4}.
    Smallest even number is 6.


    Start with 73+6=79
    R1=2,1,5,2,9. THE SET OF FACTORS + R1 IS {3+2,5+1,7+5,9+2,11+9}.
    Smallest even number not in R1 is 4. 

    79+4=83

    FAQ
    • Can this procedure be optimized further?
    Yes. By considering fewer factors such as prime numbers only instead of odd numbers. 

    • When should we include a new factor or prime as a divisor?
    When there is a new prime number whose square falls between N0 and N1, then we include that new prime as another factor. We find its R1, include it in our set and proceed as before.

     More examples to show when the above simple formula fails:
    Take a prime number 1301
    Its Sq root is approx 36

     Prime numbers less than 36:
    3,5,7,11,13,17,19,23,29,31
    R1 values of 1301 when divided by above prove numbers
    3; 1
    5: 4
    7: 1
    11: 8
    13: 12
    17: 8
    19: 10
    23: 10
    29: 4
    31: 1

     Minimum even number in the set is 6. THE SET OF FACTOR + R1 IS {3+1,5+4,7+1,11+8,13+12,17+8,19+10,23+10,29+4}.
    The next prime number after 1301 is 1301+6 = 1307.

     An interesting example is prime number 113
    The square root is approx 11.
    The R1 values and R1 + factor values are
    3; 1        3+1=4
    5: 2        5+2=7
    7: 6        7+6=13
    11: 8    11+8=19

     The minimum even number which is not in above sets {1,2,6,8}, {4,7,13,19} is 10. Adding 10 to 113 gives 123 which is divisible by 3. Hence our formula needs to be corrected. WE HAVE TO LOOK FOR THE LOWEST EVEN NUMBER WHICH IS NOT IN R1 NOR SHOULD IT BE IN THE SET OF (SUM OF R1 + n*factor) where n is any positive integer. That is, we have to look for the smallest even number from which when R1 is subtracted, the reminder is NOT a multiple of the factor; this should be true of each of the prime factors. Such an even number, that satisfies this condition, when added to N1 will give the next prime N2.

     The even number 10 is R1+3*factor where factor is 3 and hence is eliminated. The next even number is 12 which is 2 (=R1 for 5) plus two times 5 and hence also eliminated. 
    Adding 12 would give a sum which is a multiple of 5. 

     The next even number is 14 which is the final answer. Hence the next prime number is 113+14.

     When there is a small even number available in the set of R1 itself, then the other set of R1 +n*factor becomes irrelevant since these usually would be larger numbers and hence wouldn't affect the min obtained from R1.
    Let's take the example of the prime which has a relatively high Gap. Let's find out this Gap
    523            R1+n*factor (only evens upto 38)
    R1 
    3: 2            8 14 20 26
    5: 2            12 22 32
    7: 2            16 30
    11: 5          16 38
    13: 10        36
    17: 4          38
    19: 9          28

     23: 6

     Prime factors up to 23 (=sq rt of 523) have been considered. Only such cases of n are taken that result in an even value for R1+n*factor . Odd values are irrelevant. Only sums up to approx 23 are needed. I have taken till 38. Even numbers 2,4,6,10 are in the 1st column showing R1. 8,12,14,16 are in the 2nd column. Hence the least even number not in either column is 18. This the Gap is 18 and hence the next prime is 531.

     Now comes the interesting thing. The even number that when added to N1 gives N2 the next prime is the Gap. That even number, we know, is approximately the min value of R1. And hence that even number is less than the factors of N1. And the highest factor is the square root of N1. Thus, the Gap for N1 is less than the square root of N1 as mentioned in the top of the article. The Gap values, as seen in the wiki article mentioned at the top of this post, are indeed of the order of the square root of N1 or much less. 

     This is an interesting proof except for one thing. That the even number is NOT JUST the min value of R1. 
    We need to also consider the min of P = R1 + factor*n
    That P could increase the Gap to potentially substantially more than the square root of N1. If we could prove that P is inconsequential then it is proved that Gap for N1 is less than sq rt (N1). Oppermann's Conjecture abstracts this further for (the square of) any number x, whether x is a prime number or not. 

     That between x^2 and x(x+1) there exists at least one prime number where x is any integer. The distance between the two algebraic expressions = x. Meaning that there exists a prime number at a distance of Sq Rt (x^2) from x^2 which is in sync with my own conclusion. 

     If the numbers N0, N1, N2 are three consecutive primes then I have suggested that N2 - N1 and N1 - N0 are both less than sq rt(N1). Take any composite number between N0 and N1. The primes N0 or N1 exists within a distance of sq rt (N1). There could be other primes even closer. The same is true of any composite number between N1 and N2. Oppermann's Conjecture could be rephrased as : For any integer X, there exists a prime number which is less than sqrt (x) distance away. 

    Another interesting fact is that I arrived at the conclusion that Gap generally does not exceed the square root of a prime number independently. My steps have been explained in this post.


    In the attached Google spreadsheet here, in the "Merit and Maximal Gap" sheet we see empirically that the ratios of Gi/ln(Ni) and Ni/i (where Ni is the ith prime number, Gi is the the corresponding maximal Gap for Ni) are approximately equal. This is a strange result. Whether it apples to higher maximal Gaps is not known. This means that the maximal Gap G(i) is approximately about N(i)*ln(Ni)/i. Other Gaps will be even smaller. 

     If this result applies to higher primes as well, we have an ESTIMATE for the Gap since the current prime N(i), the value of i and the Gap are all known. 

     As I see for primes up to 5000, Gap that occurs most frequently is 6. Why 6? Why not 2 or any other even number? I think the reason is 6 is a multiple of 3. By adding a multiple of 3, we have already ensured that the new number is definitely not a factor of 3. Note 3 is also a multiple of 3 but 3 cannot be the Gap because 3 is odd.
     Similarly one can say that Gap of 6*5 =30 is likely to be very frequent for high primes. Similarly 30*7=210, 210*11= 2310 etc are likely to be frequently occurring Gaps. The same thing is mentioned in https://golem.ph.utexas.edu/category/2016/03/the_most_common_prime_gaps.html

     That leads to the question: Determine the first prime number that has a Gap of 210. Since 210 is a multiple of 3, 5, 7 the values across columns in the first the rows in Factor sheet is irrelevant. 210 will not be among the even numbers there. So, we need a scenario where (1) 210 does not exist but (2) all even numbers up to 208 do exist, at least in Union with the first three columns, in the first two columns across all rows. 
    Additional Reading
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    Wednesday, April 3, 2019

    Birds Of The Same Feather

    It's when likes attract that there is something beautiful in a relationship. When opposites attract it's usually a question of shopping for a need. This leads to a temporary honeymoon followed by boredom or an active engagement in other affairs (no pun intended).

    Applying this to physical relationships between people, it kinda implies that heterosexual relationship is cheaper than homosexual. This conclusion seems obviously untrue. 

    But I guess the true meaning is that a desire for someone based on a fatter wallet and / or a rounder behind (or BTSA) is a cheaper basis.
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