Number Theory

If you have taken some math tuition on number theory, you should be familiar with the concept of prime numbers. Prime numbers are those numbers which are only divisible by 1 and that number itself. Some examples of prime numbers are 2, 3,7,11 13 etc. These numbers can only be divided by themselves or 1 without producing a remainder.

Prime numbers are one of the most beautiful things in mathematics. They are also quite mysterious in some sense. For thousands of years, many great minds have worked with them to uncover some of their mysteries. The first and foremost thing that the mathematicians wanted to find about prime numbers is a definitive pattern in their distribution. For example, let us consider the following
sequences of numbers:

Sequence 1: 2, 4, 6, 8, 10.....

Sequence 2: 2, 4, 8, 16....

Sequence 3: 1, 1,2,3,5,8,13...


In all of the above 3 sequence of numbers, we can see a definitive pattern. In the first sequence, all numbers are separated by an interval of 2 i.e. we get the next number by adding 2 to the current number. All the numbers generated in this sequence will be even if we follow the same pattern, given we start with an even number. If we take the example of the second sequence, each number (apart from the first number of course) is double of the previous number i.e we generate the next number by multiplying 2 to the current number. The third sequence is known as the Fibonacci sequence and there is a definitive pattern to it as well. Can you see the pattern? Well, each number, apart from the first two numbers (i.e. 1 and 1), is the sum of the previous 2 numbers. For example, 3 is 2+1 and 13 is 8+5. As we can clearly see, there is a definite pattern. The mathematicians wanted to find the same thing for prime numbers; they wanted a pattern using which prime numbers could be generated and predicted. Although there was no lack of trying and many great minds of all time wrestled with this particular problem, till date, we have failed to find such a definitive pattern.


Leonhard Euler, one of the greatest mathematicians of all time, in fact came up with a formula for generating prime numbers. Those of you who have taken some math tuition on algebra or number theory might already be familiar with this formula. The formula goes as follows: n2 – n + 41
If we plug-in the values of n, we will indeed generate primes – well, at least in some cases. For example, when we plug-in n=1, we get 41 – prime. If we plug-in n=2, we get 43 – again a prime. If we plug-in n=3, we get 47, a prime!! In fact, this formula works all the way up to 40. The point it fails at is when n=41 when the value of the expression becomes 1763 which is not prime because this number is the product of 43 and 47 and hence devisable by both of them. As a result, although the formula shows some promise early on, it fails at a point. In fact, this formula potentially generates an infinitely many composite numbers. Although we have failed to find a general pattern which predicts the distribution of prime numbers, it is important to mention here that we have been able to find certain patterns which hold true in certain cases. For example, the Ulam spiral, the prime number chains, Gilbreath's conjecture and many others are examples where such patterns are observed.
Number pattern is a familiar topic for maths tuition students in Singapore. Though it starts from the basic primary maths tuition classes in Singapore, maths tuition students are well aware of number patterns. One of the very popular maths questions many years ago was to find the sum of numbers from 1 to 100. This question was for 12 year olds students taking the maths examination.

Maths Tuition – The Best Option for Kids Who Struggle With the Digits

Maths classes that are conducted in the schools are not always the finest options for students when it comes to getting through their exams, particularly for those who are always at daggers drawn with the subject. There are certain students who have a tendency of running away from Maths because they are simply afraid of the subject and this is what makes institutes like Miracle Learning Centre so much pertinent in today’s context.

Going back to the students’ acumen or the lack of it in grasping Maths, there is simply no reason to blame the teachers of the schools. They after all, have the responsibility of taking care of a number of students during a short period of time. Hence, they in most of the cases do not have the time to give that extra bit of time to those who need some special and focused attention. To this hapless chunk of students, maths tuition is the only feasible solution, especially for those who are really struggling to get in terms with even the elementary issues of the subject.

Here, the only point that needs to be taken into account very seriously is the type of tutorial to opt for. Indeed, the tutorial needs to be chosen very immaculately so the move does not have any adverse effect on the psyche of the student in question. Everything needs to be taken into account – right from the qualification of the tutors, to their experience and method of teaching, and last but not the least, their ability to quickly and seamless adapt to the learning style of the student in question. Experienced Maths tutors in tutorials like Miracle Learning Centre encourage students to respond to questions and also clear their doubts. There are certain students who take time to open up with their problems and issues. Tutors who conduct these maths tuition classes come up with specialized methodology of teaching to help these students divulge their problems so that they can be addressed and taken care of in a methodical way as per the mental structure and mental setup of the students. We at Miracle Learning Centre are excellent at this, as we are home to some of the most experienced and trained tutors who take engaging sessions and equip the students with appropriate knowledge in each step and this helps the students to gain confidence and the required knowledge. All these help them subsequently to appear in Maths assessments with the required self-belief and a renewed vigor. This, ultimately reflects on their mark sheets in the long run.