The Amazing Concepts

In number theory, a Practical number or Panarithmic number is a positive integer n that can represent all smaller numbers m (m < n) as the sums of distinct divisors of n. In other words, any positive integer n having the property that all smaller integers can be represented as the sums of distinct divisors of integer n is referred to as a Practical number or Panarithmic number. The Procedure Step 1:  Consider a positive integer n. Step 2:  Find out the distinct divisors of n. Step 3:  Check whether all numbers smaller than n can be represented as the sums of distinct divisors of n. Step 4:  If all numbers smaller than n can be represented as the sums of distinct divisors of n, then n is a Practical number or Panarithmic number, else not. Practical Number Sequence 1, 2, 4, 6, 8, 12, 16, 18, 20, ….., n. For Example Integer: 24 Distinct divisors of 24: 1, 2, 3, 4, 6, 8, 12. All smaller numbers such as: 5 => 2 + 3 7 => 3 + 4 9 =>...

D. R. Kaprekar (Dattatreya Ramchandra Kaprekar) introduced an amazing concept Kaprekar number in number theory. Kaprekar number is a natural number whose square when divided into two equal or non-equal parts; left and right, then the sum of both parts must be equal to the original number. None of the parts should have the value zero. The Procedure Step 1: Consider a natural number n. Step 2: Square the number n 2 . Step 3: Divide the value after square into two equal or non-equal parts: left x or x-1 digits and right y digits. Step 4: Add both left and right parts: (x + y) or (x-1 + y). Step 5: If the resultant z is equal to the original number n, then number n is a Kaprekar number, else not. For Example Number n: 9 Square the number n 2 : 9 2 => 81. Divide the value after square into two equal or non-equal parts: 8 and 1. Add both left and right parts: 8 + 1=> 9. Here the resultant 9 is equal to the original number n. Hence, 9 is a Kaprekar number. ...

Lemoine’s conjecture or Levy’s conjecture, named after Emile Lemoine and Hyman Levy, respectively, is a popular conjecture as far as the number theory is concerned. According to this conjecture, every odd positive integer greater than 5 can be expressed as the sum of an odd prime number and an even semiprime. A prime number other than 2 is called an odd prime number . A semiprime is the product of two prime numbers . As per the definition, every odd positive integer n > 5 is the sum of an odd prime p and an even semiprime 2q. Mathematically,  n = sum(p + (2q)) . Conditions Condition 1:  Integer n must be odd. Condition 2:  Integer n must be greater than 5. For Example Integer n: 7 Sum of an odd prime p and even semiprime 2q: 3 + (2 x 2) => 7. Integer n: 11 Sum of an odd prime p and even semiprime 2q: 5 + (2 x 3) => 11. Integer n: 13 Sum of an odd prime p and even semiprime 2q: 7 + (2 x 3) => 13. Integer n: 17 Sum of an odd pri...

Christian Goldbach, a Russian mathematician [March 18, 1690 – November 20, 1764], given a popular conjecture in number theory named as Goldbach’s conjecture . According to this conjecture, every even positive integer greater than 2 can be expressed as the sum of two prime numbers. As per the definition, every even positive integer n > 2 is the sum of two prime numbers p and q. Mathematically, n = sum(p, q) . Conditions Condition 1: Integer n must be even. Condition 2: Integer n must be greater than 2. For Example Integer n: 10 Sum of two prime numbers p and q: 3 + 7 => 10. Integer n: 44 Sum of two prime numbers p and q: 3 + 41 => 44. Integer n: 58 Sum of two prime numbers p and q: 5 + 53 => 58. Integer n: 90 Sum of two prime numbers p and q: 7 + 83 => 90. Integer n: 100 Sum of two prime numbers p and q: 3 + 97 => 100. H ope you have understood this amazing concept of Goldbach’s conjecture.

This post is about some amazing facts that are related to the number theory. Below is the list of these amazing facts: Fact No. 1: All four-digit palindrome numbers are divisible by 11. Example: Palindrome numbers are those numbers which remains the same while reading from left to right and right to left. 1001 is divisible by 11, and the resultant is 91. 2552 is divisible by 11, and the resultant is 232. 4664 is divisible by 11, and the resultant is 424. 7227 is divisible by 11, and the resultant is 657. 9889 is divisible by 11, and the resultant is 899. Fact No. 2: Repeat a three-digit number twice to form a six-digit number. The formed six-digit number is divisible by 7, 11 and 13. Again dividing the formed six-digit number by the multiplication resultant of 7 x 11 x 13, gives back the original three-digit number. Example: The original number: 123. Repeat the number twice to form a six-digit number: 123123. The formed six-digit number 123123 is divisib...

The number 1089 is considered a magical number as far as the number theory is concerned. Why is this number considered a magical number? The answer to this question is described in this post. The Concept Step 1: Take any three-digit positive integer with non-identical digits (not all digits are same). Step 2: Reverse the digits of the number to form mirror number Step 3: Subtract mirror number from the original number taken in Step 1 . Step 4: Calculate the difference and reverse the digits of the difference. Step 5: Add the digits of the difference to its reverse digits. The end-result is always the number 1089. For Example Number 123 Mirror number formed after reversing the digits is 321. Subtract mirror number from the original number: 123 – 321 => -(198). Difference after calculation is 198 and its reverse digits are 891. Adding digits of the difference to its reverse digits: [198 + 891] => 1089 . As you can see, the end-result is the number 1089. ...