024. More number theory

Question: `q001. There are 8 questions in this assignment.

Pick any even number--say, 28.

It is believed but not yet proven that whatever even number you pick, as long as it is at least 6, you can express it as the sum of two odd prime numbers. For example, 28 = 11 + 17.

Express 28 as a some of two prime factors in a different way. 

Your solution:

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Given Solution:

28 can be expressed as 5 + 23, both of which are prime.

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Question: `q002. The assertion that any even number greater than 4 can be expressed as a sum of two primes is called Goldbach's conjecture. Verify Goldbach's conjecture for the numbers 42 and 76.

Your solution:

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Given Solution:

42 = 23 + 19, or 13 + 29, or 11 + 31, or 5 + 37.

76 = 73 + 3, 71 + 5, 59 + 17, 53 + 23, or 29 + 47.

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Question: `q003. The proper factors of a number are the factors of that number of which are less than the number itself. For example proper factors of 12 are 1, 2, 3, 4 and 6.

List the proper factors of 18 and determine whether the sum of those proper factors is greater than, less than, or equal to 18 itself.

Your solution:

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Given Solution:

The proper factors of 18 are easily found to be 1, 2, 3, 6 and 9. When these factors are added we obtain 1 + 2 + 3 + 6 + 9 = 21. This result is greater than the original number 18.

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Question: `q004. A number is set to be abundant if the sum of its proper factors is greater than the number. If the sum of the proper factors is less than the number than the number is said to be deficient. If the number is equal to the sum of its proper factors, the number is said to be perfect.

Determine whether each of the following is abundant, deficient or perfect:

12; 26; 16; 6.

Your solution:

Confidence Assessment:

Given Solution:

The proper factors of 12 are 1, 2, 3, 4 and 6. These proper factors add up to 16, which is greater than 12. Therefore 12 is said to be abundant.

The proper factors of 26 are 1, 2, and 13. These proper factors add up to 16, which is less than 26. Therefore 26 is said to be deficient.

The proper factors of 16 are 1, 2, 4 and 8. These proper factors add up to 15, which is less than 16. Therefore 16 is said to be deficient.

The proper factors of 6 are 1, 2, and 3. These proper factors add up to 6, which is equal to the original 6. Therefore 6 is said to be perfect.

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Question: `q005. There is a perfect number between 20 and 30. Find it.

Your solution:

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Given Solution:

The numbers 23 and 29 are prime, and no prime number can be perfect (think about this for a minute and be sure you understand why).

20 has proper factors 1, 2, 4, 5 and 10, which add up to 22, so 20 is abundant and not perfect.

21 has proper factors 1, 3 and 7, which add up to 11, which make 21 deficient.

22 has proper factors 1, 2 and 11, which add up to 14, so 22 is deficient.

24 has proper factors 1, 2, 3, 4, 6, 8 and 12, which add up to 35, so 24 is abundant.

25 has proper factors 1 and 5, and is clearly deficient.

26 was seen earlier to be deficient.

27 has proper factors 1, 3 and 9, and is clearly deficient.

28 has proper factors 1, 2, 4, 7 and 14. These add up to 28. So 28 is a the perfect number we are looking for.

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Question: `q006. Why can't a prime number be perfect?

Your solution:

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Given Solution:

A prime number has only two factors, itself and 1. It therefore has only one proper factor, which is 1. Since every prime number is greater than 1,no prime number can be perfect.

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Question: `q007. 2^2 - 1 = 3, which is prime. 2^3 - 1 = 7, which is prime. 2^5 - 1 = 31, which is prime. Is it true that for any n > 1, 2^n - 1 is prime?

Your solution:

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Given Solution:

You might at first think that the above examples establish a never-ending pattern, but if you check n = 4 you find that 2^n - 1 = 2^4 - 1 = 16 - 1 = 15, which is divisible by 3 and 5 and is not prime.

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Question: `q007. Check for p = 3, then p = 5, then p = 7 to see whether the formula 2^p - 1 seems to give primes, where p stands for a prime number.

Your solution:

Confidence Assessment:

Given Solution:

2^3 - 1 = 8 - 1 = 7.

2^5 - 1 = 32 - 1 = 31.

2^7 - 1 = 128 - 1 = 127.

All these results are prime. However this doesn't prove that the formula always works. Your book will address this question.