We know that 10* 10 = 100.  

The square which are whole numbers greater than 10 until it exceeds that the given limit of this 500.

10* 10 = 100 (It is not included as the number should lie in between 100 and 500).

11 * 11 = 121.  

12 * 12 = 144.  

13 * 13 = 169

14 * 14 = 196

15* 15 = 225

16 * 16 = 256

17 * 17 = 289

18 * 18 = 256

19 * 19 = 361

20* 20 = 400.  

21 * 21 = 441

22 * 22 = 484

23 * 23 = 529 (It is not included as the number be in between 100 and 500)

The perfect square need to exit in between 100 and 500 that are 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441 and 484. There are mainly 12 number in between.

Perfect number is a positive number which is equal to the sum of proper divisors. The smallest perfect number is 6 which is the summation of three numbers that is 1,2 and 3. Some of the other perfect square numbers are 28, 496 and 8128. Discovery of these numbers are completely lost in the prehistory. In the 525 B.C Pythagoreans found that the number come up mystical properties. The tradition was completely continued by Neo-pythogorean philosopher who have already classified the number as deficient, superabundant and perfect. It is the result of sum of their divisor which is much less than or even equal to the given number. Nicomachus comes up with moral qualities to the given definition that found these kind of idea among the credence in the Christian Theologians. In the cycle of 28 days for Moon around Earth is given an example that is heavenly which is perfect. Even if it is naturally perfect, most of the example is provided by St. Augustine.

The earlier of the extant mathematical result is mainly concerned to the perfect numbers which occurs in Euclid elements that provides proposition. In these scenario, Double proposition means that the number is completely twice the preceding number that is 1, 2, 4, 8….. . The Most suitable example is 1+2+4 =7 which is a prime number. So, as a result the sum needs to be multiplied to the last number which is found to be perfect number. The formula of Euclid forces that any perfect number is obtained from the even number. In the 18^th century, Leonhard Euler who is Swiss mathematician highlighted that perfect number needs to be obtained from Euclid formula. It is not that whether there are is existence of any odd perfect number or not.

Prime number is any positive number which is considered to be greater than the value of 1 and is divisible by itself by number. The most suitable examples are 2,3,5,7,11, 13,17 and 23. The key result of number theory is called the fundamental theorem of arithmetic that states every positive integer is greater than the value 1 can be stated as the product of prime numbers. This is considered to be a unique fashion. As prime are regarded as the building blocks that is multiplicative for natural numbers.

Prime numbers are considered to be antiquity which is studied by Euclid who is a Greek Mathematician. In the given elements, Euclid provided the first proof that are infinite number of prime numbers. There are large number of formulas which are suggested for discovering the prime numbers but most of them come up with flaw. Two of most famous result that result in distribution of the prime numbers are prime number theorem and Riemann zeta function. In the late 20^th century, by the help of developed system, prime number that have millions of digit have been discovered. Similar to the efforts that is being generated in more digits of π. This kind of number theory research highlighted there is no kind of possible application of it. Till then, cryptographers have discovered how the prime numbers make use of unbreakable codes.

In the number theory, a perfect number can be stated as a positive integer which is a sum of positive divisor which is exclusive of number itself. For example, 6 comes up with divisor that is 1, 2 and, 3 (exclusive the number itself) so 6 is a perfect number. The number of divisor of the number is mainly exclusive of number which is aliquot sum. It can be defined as a perfect number which is one that is completely equal to aliquot sum. A perfect number can be defined as a number of which has a half of sum for all the positive divisor inclusive of the number itself. For example, 28 is a perfect number 1+2+4+7+14+28=56= 2 * 28.

The point is not clear till now that there are any old number or where there is existence of any perfect number that is 6, 28, 8128 and 496. The point should be noted that there are any odd number perfect number which comes up into picture through odd perfect number. This mainly comes up into picture through various obtained. In the year 1496, Jacques come to the point that rule of Euclid provides all the perfect numbers. This merely implies that odd number perfect number can exist.

All the prefect numbers that are even that they aim to exit in precise form. Odd numbers may or may not exist or even in rare cases. There are number of result for perfect number which are quite easy to move but it nevertheless impressive in nature. The sum of divisors aims to provide various kind of numbers. There are numbers where the sum stands to be much less than the number itself and it is considered to be deficient. When it is greater than the number then it is termed as abundant. A pair of given number where sum is proportional to each other with proper divisor it is defined as amicable.