Wednesday, August 13, 2008
Yes they can be happy?
Then, what is a happy number?
A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.
More formally, given a number n = n0, define a sequence n1, n2, ... where ni + 1 is the sum of the squares of the digits of ni. Then n is happy if and only if there exists i such that ni = 1.
If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.
For example, 7 is happy, as the associated sequence is:
7^2 = 49
4^2 + 9^2 = 97
9^2 + 7^2 = 130
1^2 + 3^2 + 0^2 = 10
1^2 + 0^2 = 1.
The happy numbers below 500 are
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496.
If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle.
4, 16, 37, 58, 89, 145, 42, 20, 4, ...
To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most 81m. For m = 4 and above,so any number over 1000 gets smaller under this process. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.
In the range 100 to 243, the number 199 produces the largest next value, of 163.
In the range 100 to 163, the number 159 produces the largest next value, of 107.
In the range 100 to 107, the number 107 produces the largest next value, of 50.
Happy Primes
A happy prime is a happy number that is prime. The happy primes below 500 are
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487.
All numbers, and therefore all primes, of the form 10n + 3 and 10n + 9 are happy. To see this, note that these numbers yield values of either 12 + 02 + 02 + ... + 02 + 02 + 32 = 10 → 12 + 02 = 1 or 12 + 02 + 02 + ... + 02 + 02 + 92 = 82 → 82 + 22 = 68 → 62 + 82 = 100 → 12 + 02 + 02 = 1.
Then, what is a happy number?
A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers.
More formally, given a number n = n0, define a sequence n1, n2, ... where ni + 1 is the sum of the squares of the digits of ni. Then n is happy if and only if there exists i such that ni = 1.
If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.
For example, 7 is happy, as the associated sequence is:
7^2 = 49
4^2 + 9^2 = 97
9^2 + 7^2 = 130
1^2 + 3^2 + 0^2 = 10
1^2 + 0^2 = 1.
The happy numbers below 500 are
1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496.
If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle.
4, 16, 37, 58, 89, 145, 42, 20, 4, ...
To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most 81m. For m = 4 and above,so any number over 1000 gets smaller under this process. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.
In the range 100 to 243, the number 199 produces the largest next value, of 163.
In the range 100 to 163, the number 159 produces the largest next value, of 107.
In the range 100 to 107, the number 107 produces the largest next value, of 50.
Happy Primes
A happy prime is a happy number that is prime. The happy primes below 500 are
7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487.
All numbers, and therefore all primes, of the form 10n + 3 and 10n + 9 are happy. To see this, note that these numbers yield values of either 12 + 02 + 02 + ... + 02 + 02 + 32 = 10 → 12 + 02 = 1 or 12 + 02 + 02 + ... + 02 + 02 + 92 = 82 → 82 + 22 = 68 → 62 + 82 = 100 → 12 + 02 + 02 = 1.
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