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...

A Nivenmorphic number or Harshadmorphic number is a positive integer whose sum of digits is the last digits of the number i.e. the sum of the digits terminates the number. The Concept Step 1:  Take any positive integer. Step 2:  Calculate the sum of the digits of the number. Step 3:  If the sum of the digits is the last digits of the number i.e. the sum of the digits terminates the number, then the number is a Nivenmorphic number or Harshadmorphic number, else not. For Example Number: 16218 Sum of the digits of the number: [1 + 6 + 2 + 1 + 8] => 18 Here, the sum of the digits 18 is the last digits of the number i.e.  the sum of the digits terminates the number.  Hence, the number 16218 is a Nivenmorphic number or Harshadmorphic number. Hope you have understood this amazing concept of Nivenmorphic number or Harshadmorphic number. You can try yourself using any other positive integer. In case of any issue, kindly pro...

The Steinhaus Cyclus is 145, 42, 20, 4, 16, 37, 58, 89 . The Concept Step 1: Take any natural number with four digits. Step 2: Add the squares of all the digits of the number. Step 3: Repeat this procedure of adding the squares of the digits of the number. Step 4: This procedure gives the result either number 1 or 145. Step 5: After getting the number 1 or 145, the Steinhaus Cyclus keeps on repeating. For Example Number 1234 Adding the squares of all the digits of the number 1234 [1 2 + 2 2 + 3 2 + 4 2 ] => 30 Repeat this procedure: [3 2 + 0 2 ] => 9 [9 2 ] => 81 [8 2 + 1 2 ] => 65 [6 2 + 5 2 ] => 61 [6 2 + 1 2 ] => 37 [3 2 + 7 2 ] => 58 [5 2 + 8 2 ] => 89 [8 2 + 9 2 ] => 145 [1 2 + 4 2 + 5 2 ] => 42 [4 2 + 2 2 ] => 20 [2 2 + 0 2 ] => 4 [4 2 ] => 16 [1 2 + 6 2 ] => 37 [3 2 + 7 2 ] => 58 [5 2 + 8 2 ] => 89 [8 2 + 9 2 ] => 145 : : : : : As you can see, after getting the num...

The number 222 is one of the amazing concepts as far the number theory is concerned. A detailed description of this amazing number 222 is described here. The Concept Step 1: Take any three-digit number with non-identical digits (not all digits are same). Step 2: Form more numbers from this number by using combinations. Step 3: Add all these numbers. Step 4: The final resultant after addition process is 222-times the sum of the digits of the number taken in Step 1 . For Example Number 123 Numbers formed after combinations from this number are: 123, 132, 213, 231, 312, 321 Adding all these numbers: 123 + 132 + 213 + 231 + 312 + 321 => 1332 The final resultant 1332 is 222-times the sum of the digits of the number 123. That is sum of the digits of the number 123 [1 + 2 + 3] is equal to 6, and it is 222-times the final resultant 1332 i.e. 1332 divided by 6 => 222. Hope you have understood this amazing concept of the number 222. You can try yourself using an...

Zuckerman number or Multiplication-Harshad number or Multiplication-Niven number is any positive integer b that is divisible by its product of digits (POD) and all of its digits. Zuckerman number cannot contain the digit 0. It is because if the digit 0 is present in the number, then multiplying it with other digits in the number gives the resultant 0, and no non-zero numbers are divisible by the digit 0. As per the definition: Zuckerman number b = number / product of digits Below are some examples: Number 24 Product of digits =  2 x 4 => 8 Here, number 24 is divisible by its product of digits 8, and the resultant is 3. Apart from this, number 24 is divisible by all of its digits: 2 and 4. Number 36 Product of digits = 3 x 6 => 18 Here, number 36 is divisible by its product of digits 18, and the resultant is 2. Apart from this, number 36 is divisible by all of its digits: 3 and 6. Number 115 Product of digits = 1 x 1 x 5 => 5 Here, nu...

The concept of Harshad number was given by D. R. Kaprekar, one of the great mathematicians from the country India. Harshad number or Niven number is any positive integer b that is divisible by its sum of digits (SOD). The term Niven number was coined by Ivan Morton Niven, a Canadian-American mathematician, who specializes in number theory. As per the definition: Harshad number b = number / sum of digits Below are some examples: Number 12 Sum of digits = 1 + 2 => 3 Here, number 12 is divisible by its sum of digits 3, and the resultant is 4. Number 18 Sum of digits = 1 + 8 => 9 Here, number 18 is divisible by its sum of digits 9, and the resultant is 2. Number 72 Sum of digits = 7 + 2 => 9 Here, number 72 is divisible by its sum of digits 9, and the resultant is 8. Number 110 Sum of digits = 1 + 1 + 0 => 2 Here, number 110 is divisible by its sum of digits 2, and the resultant is 55. Number 135 Sum of digits = 1 + 3 + 5 => 9 Here, number...

The 196-Problem is one of the biggest unsolved mysteries as far as the mathematics is concerned. The reason  behind this mystery is that it has not been proved yet whether 196 is a  Lychrel number or non-Lychrel  number ; a number that produces  Palindrome number  after repeatedly performing the reverse-then-add operation. As you can see, the terms – Lychrel number, non-Lychrel number, and Palindrome number have been used above. Before going into detail of The 196-Problem, it is important to know about these terms in order to understand this problem effectively. Lychrel number The name Lychrel was coined by computer scientist Wade Van Landingham. Lychrel number is a natural number that do not result in forming numeric palindromic sequence or Palindrome number even after pefroming multiple iterations  [n]  of the reverse-then-add operation. Non-Lychrel number If any natural number result in forming numeric palindromic sequence or Palindrome number afte...

Dattatreya Ramchandra Kaprekar (D. R. Kaprekar; 1905 – 1986) was an Indian mathematician who discovered several classes of natural numbers including Kaprekar Constant. The number  495  is called Kaprekar Constant for three-digit number. Why is the number 495 called Kaprekar Constant? In 1949, D. R. Kaprekar discovered an interesting property of the number 495 through the mathematical calculations. D. R. Kaprekar observed that when any three-digit number of non-identical digits (not all digits are same) is arranged in two parts – highest number and lowest number, and difference is calculated using subtraction method, then the end-result will be always 495. If the process is continued even after getting the end-result 495, then the end-result will be always 495 at each Iteration, therefore the number 495 is known as Kaprekar Constant. Steps to Follow Take any three-digit number with different digits. Form the highest number and lowest number from this three-digit number. Fi...