# 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 = ____ + ____ + ____

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