Find a perfect number other than 6 and less then 30
Answers
Answer:
What is a perfect number?
A perfect number is a whole number, an integer greater than zero; and
when you add up all of the factors less than that number, you get that number.
Examples:
The factors of 6 are 1, 2, 3 and 6.
1 + 2 + 3 = 6
The factors of 28 are 1, 2, 4, 7, 14 and 28.
1 + 2 + 4 + 7 + 14 = 28
The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496
The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128. I'll let you add them up.
According to The Merriam-Webster Dictionary, the term was first used in the fourteenth century. The Grolier Multimedia Encyclopedia says that perfect numbers are "another example of Greek progress in number theory," and credits the Pythagoreans for coining the term "perfect." If you are interested in learning more about "perfect" numbers, you should also read up about "Mersenne" prime numbers because they are closely related.
The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic.
I love challenges, so I wrote a program to use trial-and-error, starting at 1 and trying every whole number going up, looking for perfect numbers. As it ran, I eventually noticed that every perfect number had a higher power of two than the prior number, so I made on-the-fly changes so that the computer would try only multiples of higher powers of two. I eventually found a total of five numbers by this trial and error method. That program eventually ran past a billion before I aborted it.
6 = 1+2+3
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
8,128 = 1+2+4+8+16+32+64+127+254+508+1016+2032+4064
33,550,336 = 1+2+4+8+16+32+64+128+256+512+1024+2048+4096
+8191+16382+32764+65528+131056+262112+524224
+1048448+2096896+4193792+8387584+16775168
Do you see a pattern in the above numbers? There are all of the powers of 2 from 1 up to a certain number, and then a prime number that is equal to DOUBLE the last power of two, minus 1.
1, 2 ------> 2*2-1 = 3
1, 2, 4 ------> 4*2-1 = 7
1, 2, 4, 8, 16 ------> 16*2-1 = 31
1, 2, 4, 8, 16, 32, 64 ------> 64*2-1=127
1,2,4,8,16,32,64,128,256,512,1024,2048,4096 ------> 4096*2-1 = 8191
The rest of the factors are each power of two TIMES that prime number.
So, our first five perfect numbers are
2 * 3 = 6
4 * 7 = 28
16 * 31 = 496
64 * 127 = 8,128
4096 * 8191 = 33,550,336
Can we turn this pattern of
(Power of Two) * (Double that Power - 1)
into a formula? Yes, we can. Let's call the prime number (2^n-1). In the above examples, we'd have (2^2-1), (2^3-1), (2^5-1), (2^7-1), and (2^13-1). This would make the other number 2^(n-1), or 2^1, 2^2, 2^4, 2^6, and 2^12.
Our formula then would be
Perfect Number = 2^(n-1) * (2^n - 1)
After reading up on how known perfect numbers relate to Mersenne Prime Numbers, I wrote another program, this time to find Mersenne prime numbers and perfect numbers. This program found eight numbers, the five above and three new numbers, which I am not going to factor.
(2^16) * (2^17 - 1) = 8,589,869,056
(2^18) * (2^19 - 1) = 137,438,691,328
(2^30) * (2^31 - 1) = 2,305,843,008,139,952,128
The two programs that I wrote are included at the end of this page.
How many perfect numbers are there?
We do not know how many perfect numbers there are. We do know that there are an infinite number of prime numbers, which means there is a very high chance that there are an infinite number of perfect numbers. This is because there is a strong link between perfect numbers and a certain kind of prime number (the Mersenne primes).
Step-by-step explanation:
what is answer give me 6 and 28