Goldbach Weak Conjecture: All odd integers greater than 7 are the sum of three odd primes
Goldbach Weak Conjecture is that all odd integers greater than 7 are the sum of three odd primes.
FIRST 28 PRIME NUMBERS, NOT INCLUDING 2: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109
My first challenge for the children was to find one solution for each by using an algorithm: starting with 3 + 3 + 3 = 9, the next odd sum of 11 would require adding two to one of the 3s to make 5 + 3 + 3 = 11. Then add two to the 5 or the 3 to get 7 + 3 + 3 = 13. Now we have to be careful — if we add two to the 7 we get 9 which is not a prime number, so we can only add two to one of the 3s, so 7 + 5 + 3 = 15; then continue until you get 7 + 7 + 7 = 21. Now you have hit a roadblock since adding two to any of the 7s will give you a non-prime 9. Now, for the sum of 23, look at the list of prime numbers and notice that 19 will require 4 more which cannot work with odd primes, so we go to 17 + 3 + 3 = 23. Then continue the original algorithm of adding two to any of the primes if it will generate another prime. Of course, there are many solutions to these numbers. This brings us to the next challenge after the children reach 109.
On the second page, I chose random odd sums of 11, 17, 35, 53, 111, 199, and 287. 11 has only one solution: 5 + 3 + 3 = 11; 17 has three solutions; 35 has 7 solutions; 53 has 16 solutions; 111 has 36 solutions; 199 has 93 solutions; and 287 has a whopping 175 solutions, all of which I put on page 4 of the pdf. I recommended a best practice for this challenge: start with the largest prime number to add to the smallest primes, then look at the two smaller primes and use Goldbach’s strong conjecture (any even number is the sum of exactly two primes) and find all of the combinations of pairs. If the children find a solutions I did not find, that will be welcomed with open arms and Mathlete Dollars. Try your best.
9 = ____ + ____ + ____
11 = ____ + ____ + ____
13 = ____ + ____ + ____
15 = ____ + ____ + ____
17 = ____ + ____ + ____
19 = ____ + ____ + ____
21 = ____ + ____ + ____
23 = ____ + ____ + ____
25 = ____ + ____ + ____
27 = ____ + ____ + ____
29 = ____ + ____ + ____
31 = ____ + ____ + ____
33 = ____ + ____ + ____
35 = ____ + ____ + ____
37 = ____ + ____ + ____
39 = ____ + ____ + ____
41 = ____ + ____ + ____
43 = ____ + ____ + ____
45 = ____ + ____ + ____
47 = ____ + ____ + ____
49 = ____ + ____ + ____
51 = ____ + ____ + ____
53 = ____ + ____ + ____
55 = ____ + ____ + ____
57 = ____ + ____ + ____
59 = ____ + ____ + ____
61 = ____ + ____ + ____
63 = ____ + ____ + ____
65 = ____ + ____ + ____
67 = ____ + ____ + ____
69 = ____ + ____ + ____
71 = ____ + ____ + ____
73 = ____ + ____ + ____
75 = ____ + ____ + ____
77 = ____ + ____ + ____
79 = ____ + ____ + ____
81 = ____ + ____ + ____
83 = ____ + ____ + ____
85 = ____ + ____ + ____
87 = ____ + ____ + ____
89 = ____ + ____ + ____
91 = ____ + ____ + ____
93 = ____ + ____ + ____
95 = ____ + ____ + ____
97 = ____ + ____ + ____
99 = ____ + ____ + ____
101 = ____ + ____ + ____
103 = ____ + ____ + ____
105 = ____ + ____ + ____
107 = ____ + ____ + ____
109 = ____ + ____ + ____
| Attachment | Size |
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| Goldbach_Weak_Conjecture_1.pdf | 259.95 KB |
