The Amazing Concepts

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...

This post is about the number 1.6 that forms the base of the mathematical concepts Fibonacci sequence and the Golden ratio. For better understanding of this amazing number 1.6, it is important to know about Fibonacci sequence and the Golden ratio. Golden Ratio The  Golden ratio is the ratio of a line segment cut into two pieces of distinct lengths such that the ratio of the longer segment to the shorter segment is equal to the whole segment to that of the longer segment. It is denoted by the symbol Phi [ø]. Suppose there is a line segment AB, and C is any point that divides the line segment AB into two pieces of distinct lengths such that AC is greater than CB. Then, as per the definition, [ø] = AC/CB = AB/AC. The value of the Golden ratio [ø] is 1.6 1803398875 . As you can see, the value of the Golden ratio starts with the number 1.6. For more information on the Golden ratio, see https://theamazingconcepts.blogspot.com/2021/07/the-golden-ratio-key-to-amazing-design....

The number 2997 is considered the mystical number because repetition of multiplication and addition leads to the number 2997 in no more than four iterations. Steps to Follow Step 1: Take any three-digit number. Step 2: Multiply each digit by 111. Step 3: Add the values obtained after multiplying each digit by 111. The end-result is the number 2997 [First Iteration] . Step 4: If the end-result is other than the number 2997, then again follow the process from Steps 1 – 3. But in this case, the number to consider will be the result obtained in Step 3. Step 5: Multiply each digit by 111. Step 6: Add the values obtained after multiplying each digit by 111. The end-result is the number 2997 [Second Iteration] . Step 7: If the end-result is other than the number 2997, then again follow the process from Steps 1 – 3. But in this case, the number to consider will be the result obtained in Step 6. Step 8: Multiply each digit by 111. Step 9: Add the values obtained after multip...

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. ...

The 3n + 1 Problem, also known as the 3n + 1 Conjecture or Collatz Problem or Collatz Conjecture, is that if any positive integer  n  is iteratively operated using two simple rules:  Addition  and  Division , then the end-result always leads to the number 1. The Rules Rule 1 [Addition]:  If the number n is odd, then triple it and add 1. Rule 2 [Division]:  If the number n is even, then divide it by 2. Steps to Follow Step 1:  Take any positive integer n. Step 2:  Check whether the integer is odd or even. Step 3:  If the integer is odd, then triple it and add 1. Step 4:  If the integer is even, then divide it by 2. Step 5:  Repeat this procedure. Step 6:  The end-result always leads to the number 1. For Example Integer n = 5 5 is an odd number, so triple it and add 1. The value of n = 16. 16 is an even number, so divide it by 2. The value of n = 8. 8 is an even number, so divide it by 2. The va...