Let's take a peek into the world of 5 digit squares and you will be surprised on some of the characteristics and patterns that is in play
I have chosen two numbers 59049 and 65536 since they are multiples of the two smallest prime numbers - 2 and 3!
59049 is 3 ^ 10 and 65536 is 2 ^ 16
a) 59049
- 243 ^ 2 or 9 ^ 5
- One of the anagram of 59049 is the number itself
- Last 2 digits is a perfect square
- Digits 4 and 5 are perfect squares as well!
- 9049 (last 4 digits) is a prime number
Now for the fascinating part - Replace the first digit (5) with numbers from 1 to 9 and we would notice that none of them are prime and at the same time product of 2 or 3 prime numbers
- 19049 is 43 * 443
- 29049 is 3 * 23 * 421
- 39049 is 17 * 2297
- 49049 is 7^ 3 * 11 * 13
- 69049 is 29 * 2381
- 79049 is 137 * 577
- 89049 is 3 * 29683
- 99049 is 37 * 2677
b) 65536
- Well know number in computer parlance
- It has so many factors of multiples of 2 - 256 ^ 2 or 16 ^ 4 or 4 ^ 8 or 2 ^ 16
- Its immediate neighbour 65537 is a prime number though!
- Again one of the anagram of 65536 is the number itself
- Last 2 digits again is a perfect square
Contrast to 59049, this is an even square but look at the following combinations
- 655 + 36 is 691 which is a prime
- 65 + 536 is 601 which is again a prime
- 6553 + 6 is 6559 which is a product of two prime numbers 7 and 937
Further let's add 59049 and 65536 - the end result is 124585
- 124585 is product of two prime numbers - 5 and 24917
- 124 + 585 is a prime number
- 12 + 4585 is a prime number
Last but not the least the last two digits of both numbers are both perfect squares (49 ad 36) and their difference is 13 which is a prime number. The sum of the square roots is also 13!
Square root of 59049 is 243 and that of 65536 is 256 - Guess what their difference is also 13!
And we can go on and on!
I have chosen two numbers 59049 and 65536 since they are multiples of the two smallest prime numbers - 2 and 3!
59049 is 3 ^ 10 and 65536 is 2 ^ 16
a) 59049
- 243 ^ 2 or 9 ^ 5
- One of the anagram of 59049 is the number itself
- Last 2 digits is a perfect square
- Digits 4 and 5 are perfect squares as well!
- 9049 (last 4 digits) is a prime number
Now for the fascinating part - Replace the first digit (5) with numbers from 1 to 9 and we would notice that none of them are prime and at the same time product of 2 or 3 prime numbers
- 19049 is 43 * 443
- 29049 is 3 * 23 * 421
- 39049 is 17 * 2297
- 49049 is 7^ 3 * 11 * 13
- 69049 is 29 * 2381
- 79049 is 137 * 577
- 89049 is 3 * 29683
- 99049 is 37 * 2677
b) 65536
- Well know number in computer parlance
- It has so many factors of multiples of 2 - 256 ^ 2 or 16 ^ 4 or 4 ^ 8 or 2 ^ 16
- Its immediate neighbour 65537 is a prime number though!
- Again one of the anagram of 65536 is the number itself
- Last 2 digits again is a perfect square
Contrast to 59049, this is an even square but look at the following combinations
- 655 + 36 is 691 which is a prime
- 65 + 536 is 601 which is again a prime
- 6553 + 6 is 6559 which is a product of two prime numbers 7 and 937
Further let's add 59049 and 65536 - the end result is 124585
- 124585 is product of two prime numbers - 5 and 24917
- 124 + 585 is a prime number
- 12 + 4585 is a prime number
Last but not the least the last two digits of both numbers are both perfect squares (49 ad 36) and their difference is 13 which is a prime number. The sum of the square roots is also 13!
Square root of 59049 is 243 and that of 65536 is 256 - Guess what their difference is also 13!
And we can go on and on!
