Square Numbers

Natural numbers are numbers that are used for counting. Among the natural numbers there are some numbers called power numbers. They are nothing but natural numbers which can be written as the product of a number with itself two or more times. Number 1 is multiplied with itself any number of times would give only 1. Therefore, 1 is a trivial power number. Power number is also understood as a number n with the equation x^k=n that has the integer solution of for some positive integer k>1.

Number 2 is the number that follows number 1 among the natural numbers. We can easily see that 2 cannot be written as product of identical numbers. Next natural number 3 also has the similar property. Both 2 and 3 are also the first two prime numbers. A prime number is a number that cannot be written as the product two distinct numbers other than 1 and itself.

Power Numbers

But the next natural number 4 is a power number. It is the product of 2 with itself. In connection with the concept of prime numbers, we can say that a power number is a composite number with identical factors. By inspection we can see that among the single digit numbers, only 1, 4, 8 and 9 are power numbers. Power numbers are very rare among the natural numbers. There are only 8 two-digit power numbers and 28 three-digit power numbers.  Among the entire 9000 four digit numbers there are only 84 power numbers. Strange though, the percentage of power numbers below is very meagre for large values of n. For example, there are only .366% of power numbers among all numbers with at most five digits.

Range	Number	%	Range	Number	%
1-9	4	44.44	1-9	4	44.44
10-99	8	8.89	1-99	12	12.12
100-999	28	3.11	1-999	40	4.4
1000-9999	84	.93	1-9999	124	1.28
10000-99999	242	.269	1-99999	366	.366

Square Numbers

Among the single digit power numbers 1, 4 and 9 are called square numbers. A square number is the product of an integer with itself.  The term ‘square’ is attributed to these numbers because of the fact that it is the area of a square with an integer side length.  Number 8 is a cube number because it is the product of 2 three times.

Square Number and the last digit

Let us see the some of the properties of square numbers. First eleven square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 ... These square numbers help us in observing a salient feature of square numbers. That is, a square number will never end in 2, 3, 7 or 8. This is a fundamental property of the square numbers.

Among the first 10 square numbers, it can be observed that square of both 1 and 9 end in 1. Similarly squares of 2 and 8 end in 4, squares of 3 and 7 end in 9 and that of 4 and 6 end in 6. It is interesting to note that 1+9=2+8=3+7=4+6=10. Hence, we can conclude that square of a number of the form 10k∓n, 1≤n≤4, ends in 1, 4, 9 and 6 respectively whereas the square of a number that ends in 0 or5 will always end in 0 and5, respectively. It is interesting to note that among these end digits 5 and 6 are not square numbers whereas all others are.

Square Numbers and the last two digits

Now, let us look at the properties of the last two digits of a square number. Though 1, 5, 6 and 9 can be the end digits of square numbers, if they are repeated, then they cannot be the end digits of square numbers. That is to say that only 00 and 44 can be the last two digits of square numbers. Among the first 100 numbers there are exactly 22 numbers that can be the end digits of square numbers. They are 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49, 56, 61, 64, 69, 76, 81, 84, 89 and 96.

There are some interesting properties for these numbers. The numbers 00, 16, 44 and 69 in the reverse order also can be the end digits of square numbers. Sum of two single digit numbers can have any value from 0 to 18. But, the sum of the last two digits of square numbers can never be 2, 14, 16 and 18. Hence, similar to that of single digit ends of square numbers, we have a result.

Given below are the types of numbers whose squares have particular end digits. Here k is any nonnegative integer.

n	Last digits of n2		n	Last digits of n2
10k	00		10k	00
10k∓5	25		10k∓5	25
50k∓1	01		50k∓1	01
50k∓2	04		50k∓2	04
50k∓3	09		50k∓3	09
50k∓4	16		50k∓4	16
50k∓11	21		50k∓6	36
50k∓18	24		50k∓7	49
50k∓23	29		50k∓8	64
50k∓6	36		50k∓9	81
50k∓21	41		50k∓11	21
50k∓12	44		50k∓12	44
50k∓7	49		50k∓13	69
50k∓16	56		50k∓14	96
50k∓19	61		50k∓16	56
50k∓8	64		50k∓17	89
50k∓13	69		50k∓18	24
50k∓24	76		50k∓19	61
50k∓9	81		50k∓21	41
50k∓22	84		50k∓22	84
50k∓17	89		50k∓23	29
50k∓14	96		50k∓24	76

Algorithm to find the square

There is an Indian short cut to find the square of a number.  Though the algorithm looks tedious and lengthy, we can write the square in a line, if we master the algorithm.

Let d_kd_k-1...d₄d₃d₂d₁ be a k-digit number.

Step 1: Take the square of the last digit i.e., d₁. Last digit of it is the last digit of the square.  Keep the carryover, if any.

Step 2: Multiply d₂ with d₁ and take twice of it and add the carryover from the previous step. The last digit of this number is the last but one digit of the square. Keep the carryover, if any.

Step 3: Take the square of d₂. Multiply d₃ with d₁ and take twice of it. Add these two sums with the carryover from Step 3. The last digit of this number is the e last but two digit of the square. Keep the carryover, if any.

Step 4: Multiply d₄ with d₁, d₃ with d₂ and take twice of both and add the carryover from the previous step. The last digit of this number is the last but three digit of the square. Keep the carryover, if any.

Continuing in this manner, in 2k-1 steps, we get the required square.

Triangular Numbers and Square Numbers

Square numbers have a fascinating link to triangular numbers. A triangular number is the sum of first n consecutive natural numbers. They are 1, 3, 6, 10, 15 ...  It is striking to note that any square number is the sum of two consecutive triangular numbers i.e., 1=1+0, 4=3+1,  9=6+3, 16=10+6, ... This leads us to conclude that twice the sum of first n natural numbers added to n+1 is a square number. i.e., if T_n is the n^th triangular number, then 2T_n+ (n+1) and 2T_n-n are square numbers.

Parity of Numbers and Square Numbers

Alas! We have another result. Difference of two consecutive square numbers is always an odd number.  Therefore, we can have the sequence of odd numbers as the difference of square numbers viz., 1-0,  4-1, 9-4, 16-9, 25-16, ... In general, the n^th odd number, say, O_n=n²-(n-1)².

Similarly, even numbers also can be found from square numbers. The parity of square numbers is same as the parity of natural numbers. Hence, the difference between every pair of alternate square numbers is as an even number. Moreover, they are all multiples of 4. Halving them we get the sequence of even numbers. Quartering them we get the natural numbers. That is to say that for any natural number n, (n+1)²-(n-1)²=2²n.  Therefore, n= [(n+1)²-(n-1)²]/2².

Prime Numbers and Square Numbers

For any given number, we know that the largest non-trivial factor it can have is its square root. Hence, in finding the factors of a number one needs to check up to its square root only. Are square numbers some way helpful to know about the distribution of prime numbers? It is easy to see that for any given n, the number of prime numbers less than n is much larger than the number of square numbers. We have seen earlier square numbers alternates parity. This gives us some clue. Take only the even square numbers. Its immediate number is likely to be a prime number.  That is, the sequence, a_n= (2n)²+1, where n does not end in 1, 4, 6 or 9, has plenty of prime numbers.  Some of the initial members of the sequence are 17, 37, 101, 197, 257, 401, 577, 677, 901, 1157, 1297, 1601,  ... Some remembrances of the sequences of Mersenne Primes and Fermat Primes.

For training in mathematics research, the collection of square numbers is a good area. Following questions are worth searchable for beginners in number theory research. Is the number of k-digit powers numbers always an even number? Is there a better connection between square numbers and prime numbers? What are the properties of the last k-digits of power (square) numbers? Various proofs for such properties would really be a hard test. How far the properties of power numbers help us in solving the integer factorization problem? To sum up, one can escalate one’s mathematical powers by doing research on power numbers and can become a square personality mathematically.

The Curious Marriage of Fibonacci Numbers and Surnames

Fibonacci Number is a number from the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ..... This sequence is a very remarkable sequence of numbers which has far and wide consequences. An Italian mathematician Leonardo Pisano alias Fibonacci is credited to have studied about this sequence and documented for the first time, after observing rabbit breeding properties.

What has Fibonacci Number to do with surnames? This observation was instigated by a recent reading of the male honeybees and their ancestry. A female honeybee is born from a fertilized egg whereas a male honeybee is from an unfertilized egg. Hence a male honey bee has only one parent. If we trace the ancestry of a male honeybee, we get the number of ancestors in each level as 1, 2, 3, 5, 8, ... . This is nothing but a Fibonacci Sequence.

Let us come to the case of surnames. Conventionally a child can inherit the surname of its father or mother.  A girl after her marriage can take the surname of her husband. However, a boy seldom takes the surname of his wife after marriage. This observation helps us to count the number of possible surnames of a boy and a girl.

A married woman can either retain her surname or accept that of her husband. Hence, a married woman has three possible surnames. This is pictorially represented below:

At any level, we could see that the number of possible surnames at any level is a Fibonacci Number. It is not just that! Spot anyone, man or woman, in this diagram and count from there: we still get a Fibonacci Sequence as a subsequence. Isn't it curious?

Propositional Logic in Chapter 8 of the Gospel According to Saint John

A proposition in Logic is a universal declarative sentence that is either true or false.  There are simple propositions formed exactly of a single sentence and compound propositions formed by connecting one or more simple propositions.  Most famous connectives used to create compound propositions are NOT, AND, OR, IF-THEN and IF AND ONLY IF.

Applying logical rules in writing is a powerful and effective method to convey the intended message. John the evangelist uses it effectively in his writings.  He being the theologian of the first century was influenced by the Greek Philosophy is a well known fact. Presentation of Jesus as the logos is the most famous of John’s contributions.  John creatively uses logical rules to establish who Jesus was and Jesus’ equality with God.

In chapter 8 of the gospel according to John the Apostle, there are plenty of logical constructions. John uses repeatedly ‘implication’ and its contrapositive. Here are some of them. (The verse number is mentioned at the beginning.  Verses referred here are mostly taken from http://www.catholic.org/bible.)

4. Law of Moses: If a woman was caught in the very act of committing adultery, then she is to be stoned.

7 Jesus: If there is anyone who is guiltless, then s/he can stone the guilty.

10 Jesus: If no one has condemned you, then I do not condemn you.

12 Jesus:  I am the light of the world; if anyone who follows me then s/he will not be walking in the dark and will have the light of life.

14 Jesus: If one knows where one comes from and where one is going, then one can be testifying on one’s own behalf.

17 Law: If the testimony is of two witnesses, then it is true.

19 Jesus: If one knows Jesus, then one knows the Father.

20 John: If one’s hour has come, then one will be arrested. No one arrested him, because his hour had not yet come.

24 Jesus: If you do not believe that I am He, then you will die in your sins.

28 Jesus: When you have lifted up the Son of man, then you will know that I am He and that I do nothing of my own accord. What I say is what the Father has taught me.

31, 32 Jesus: If you make my word your home, then you will indeed be my disciples and you will come to know the truth, and the truth will set you free.

34 Jesus: Everyone who commits sin is a slave.

36 Jesus: If the Son sets you free, then you will indeed be free.

39 Jesus: If you are Abraham's children, then you will do as Abraham did.

40, 41 Jesus: If you do not do what Abraham did, then you are not Abraham’s children.  (Contrapositive  of 39)

42 Jesus: If God were your father, you would love me, since I have my origin in God and have come from him and ; I did not come of my own accord, but he sent me.

43 Jesus: You do not understand what I say, because you cannot bear to listen to my words. Contrapositive of “if you bear to listen to my words, then you do understand what I say.”

47 Jesus: If someone comes from God, then s/he listens to the words of God.

48 Jesus: You do not listen to the words of God. Hence, you are not from God. (Contrapositive of 47)

50 Jesus: There is someone who does seek his/her own glory and is the judge of it.

51 Jesus:  If you keeps my word then you will never see death.

52 Jews: Abraham is dead, Prophets are dead. Does it mean that they did not keep the word of God?
54 Jesus: If one is to seek one's own glory, then one's glory would be worth nothing.

The Cross of Christ

Open Air Granite Cross at the CMI Monastery, Elthuruth, Kerala

The Cross of Christ is the way of life.
Cross of life is the Way of Christ.
The way of life is the Cross of Christ.
The Cross of Christ saves your life.

When deeds cross with the laws of the world,
You are nailed on the cross.
Nailed on the Cross,
Your laws are not of the world.

The Cross is the way to Christ.
The Cross is the way of Christ.
The Cross is the way to heaven.
The Cross is the way of heaven.

It is the only way,
Between heaven and world.
When on the cross, you are raised from world.
If you want to be raised from the world,
Be on the Cross.

On the cross, you are condemned by the world.
If you want to be pleased by God,
Be on the Cross.

Cross was for the trouble makers.
But the Cross solves all the troubles.
Cross was for humiliation.
But the Cross annihilates all humiliation.

Abel Prize 2014 for Yakov Grigorevich Sinai

The Abel Prize for 2014 has been awarded to Russian mathematician Yakov Grigorevich Sinai on 26 March by Nils Chr. Stenseth, President of the Norwegian Academy of Science and Letters at Oslo.  He is the thirteenth recipient of this highly coveted prize which is the equivalent of the Nobel Prize for Mathematics.  The Abel Prize was presented to the laureate by H.R.H. Crown Prince Haakon at the award ceremony in the University Aula, Oslo, Norway on May 20, 2014.

Yakov Sinai is a stalwart in the theory of dynamical systems, mathematical physics and probability theory. He is an alumnus of Moscow State University. After leaving Moscow State University in 1993, he became a professor of Mathematics, at the Princeton University.

In addition, Sinai is also associated with the Landau Institute of Theoretical Physics and Russian Academy of Sciences. He has written more than 250 research articles and has supervised more than 50 PhD students.

Ragni Piene, chair of the Abel committee, lauded Sinai’s seminal contributions in shaping the modern metric theory of dynamical systems. World renowned mathematician Jordan Ellenberg gave a popular science presentation of Sinai's work. Besides Piene of University of Oslo, the Committee comprises renowned mathematicians namely Cédric Villani, Institut Henri Poincaré and Université de Lyon, France, Maria J. Esteban, Ceremade, Paris, France, Stanislav Smirnov, Section of Mathematics, University of Geneva, Switzerland and Gang Tian Mathematics Department Princeton University, US, and School of Mathematical Sciences, Beijing University, China.

The Abel Prize was instituted in memory of Niels Henrik Abel (1802-1829) who is acknowledged as the greatest son of Norway and considered among the pioneers of modern Algebra. While the Indian mathematical prodigy Srinivasa Ramanujan, died young of pneumonia, similarly Abel also passed away at an equally young age of tuberculosis. Coincidentally both these geniuses suffered a life of poverty.    

The Niels Henrik Abel Memorial Fund was established in 2002, in the bicentenary year of Abel’s birth to award the Abel Prize for outstanding scientific work in the field of mathematics. The six million Norwegian Kroner prize amount (INR six crore) was intended to strengthen and inspire teaching as well as scientific efforts. It re-established the uniqueness and centrality of Mathematics among all forms of knowledge -- but was excluded from the elite list of Nobel Prize subjects.

The Abel Prize was awarded for the first time in June 2003 to the French mathematician Jean-Pierre Serre. In 2004 and 2008 the honour was shared by two people each. Interestingly reputed  Indian mathematician Srinivasa S. R. Varadhan of Courant Institute of Mathematical Sciences, New York won the Prize in 2007 for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviations.

The award committee was highly appreciative of Sinai’s identification of umbilical relation between order and chaos. The committee noted Sinai’s contributions in the areas of Ergodic theory and Statistical Mechanics.  The Ergodic theory studies the tendency of a system to explore all of its available states according to certain time statistics whereas statistical mechanics explores the behavior of systems composed of a very large number of particles, such as molecules in a gas.

Sinai along with his PhD advisor Andrey Kolmogorov, framed the Kolmogorov–Sinai entropy, a mathematical foundation for determining the number that defines the complexity of a given dynamical system such as weather, the motion of planets, economic systems etc.

As a forerunner in Ergodic theory, Sinai proved the first ergodicity theorems for scattering billiards in the style of Boltzmann. He constructed Markov partitions for systems defined by iterations of Anosov diffeomorphisms which led to a series of outstanding works showing the power of symbolic dynamics to describe various classes of mixing systems. He is well known for Sinai-Ruelle-Browen measures, Sinai’s walks, Pirogov–Sinai theory, the stochastic Burgers equation of E–Khanin–Mazel–Sinai, the Bleher–Sinai renormalization group theory.

Professor Sinai has trained and influenced a generation of leading specialists in his research fields. Much of his research has become a standard toolbox for mathematical physicists. His works had and continue to have a broad and profound impact on Mathematics and Physics, as well as on the ever-fruitful interaction of these two fields. He says to Ellenberg: “Mathematics and physics must go together as horse and carriage.” In his popular article, “Mathematicians and Physicists = Cats and Dogs?,” Prof. Sinai had written that for him theoretical physics played the same role as experimental physics played for a physicist.The journal Nature calls him a chaos-theory pioneer who developed fundamental tools for the study of unpredictable phenomena.

Professor Sinai has won almost all great awards in Mathematics which includes Boltzmann Medal (1986), Dannie Heineman Prize (1990), Dirac Prize (1992), Wolf Prize (1997), Nemmers Prize (2002) and Henri Poincaré Prize (2009). On his 75^th birthday, the Moscow Mathematical Journal named him one of the greatest mathematicians of our time. It adds, ”Professor Sinai’s mere presence at a seminar or at a conference makes scientific life brighter and more exciting.”