This number is,
inter alia, a sum of 3 consecutive primes and can also be represented as a sum of 3 non-zero squares (not necessarily distinct) in 8 different ways.
Find the smallest number fitting the above description.
What else can be added re this number?
Part 1:
----------
The smallest number that fits the description is 689.
Sum (227, 229, 233) = 689, where 227, 229, and 233 are consecutive primes.
Part 2:
----------
Some selected properties of the number 689 are furnished hereunder as follows:
--> Property 1:
689 can be represented as sum of 3 nonzero squarea (not necessarily distinct ) in 8 different ways as follows:
o 2^2+3^3+26^2 =689
o 2^2+ 18^2 + 19^2 =689
o 3^2+14^2+22^2 =689
o 4^2+12^2 +23^2 = 689
o 6^2+13^2+22^2 =689
o 7^2+8^2+24^2=689
o 8^2+15^2+ 20^2 = 689
o 12^2+16^2+17^2= 689
o 13^2+14^2+18^2=689
--> Property 2:
689 is a strobogrammatic number, since it remains unaltered when read upside down.
--> Property 3:
The sum of the prime factors {13, 53} of 689 is 66, which is a palindrome.
--> Property 4:
689 is digitally balanced in binary since its base 2 representation 1010110001 contains an equal number (5) of 0s and 1s.
--> Property 5:
If can be separated into concatenation of 68 and 9, whereby we observe that 68+9=77, which is a palindrome.
--> Property 6:
689 is equal to 373 in base 14, a non-trivial palindrome.
--> Property 7:
689 can be written as the sum of squares of distinct positive integers in two different ways as follows:
17^2+20^2= 8^2+25^2 = 689
Edited on August 1, 2022, 7:32 am
Puzzle Thoughts
