Goldbach Strong Conjecture (Even numbers > 2 = the sum of exactly two primes)

Every even integer greater than two can be expressed as a sum of two primes, is one of the most enduring unsolved problems of mathematics. The conjecture was first made in a letter Christian Goldbach (from Prussia now Russia) wrote to the Swiss mathematician Leonhard Euler on June 7, 1742 (Goldbach went to Moscow in 1728 to be a tutor to Tsar Peter II) . In his response on June 30 of the same year, Euler said: 

“Every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it.” This has remained as one of the best known unsolved problems of mathematics to this day and is referred to as the ‘strong’ Goldbach conjecture.

The number of solutions for any even number is called “partitions.” For example, the number 6 can only have one solution at 3 + 3. 8 has one partition or solution at 3 + 5. 10 has two partitions at 3 + 7 and 5 + 5. Some notable challenges to try:

1. 100 has 6 partitions. Can you find them all? 100 =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ = ____ + ____  =  ____ + ____

1. 90 has the most partitions under 100. Can you find all 9? 90 =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ = ____ + ____  = 

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ 

3.   210 has 19 partitions. Can you find them all?  210 =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ = ____ + ____  =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ = ____ + ____  =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____ = ____ + ____  =

____ + ____  = ____ + ____ = ____ + ____  = ____ + ____

Pages 6, 7, and 8 of the attached pdf have challenges for the students to find all of the solutions for the even numbers from 16 through 100 with the number of solutions listed.

I had the K-1st graders first add prime numbers on pages 3 and 4 of the pdf. Then for some of them, I had them try to find the Goldbach pairs for even numbers 16 through 32.

CAN YOU FIND A PATTERN IN PRIME NUMBERS? HERE ARE SOME ATTEMPTED PATTERNS:

1. 3 31 331 3,331   33,331      333,331     3,333,331 33,333,331  are all prime

BUT then I tried 333,333,331 and found that it equalled 17 x 19,607,843 “aw-shucks”

2.   61 661 6,661   are all prime BUT 66,661 = 7 x 9523

3.   59 659 6,659 are all prime BUT 66,659 = 191 x 349

4.   2^2 - 1 = 3    2^3 - 1 = 7    2^5 - 1 = 31    2^7 - 1 = 127, BUT 2^11 - 1 = 2,047=23 x 89

These 2^n -1 primes (one less than a power of two) are called Mersenne Primes discovered 350 years ago (51 are now known. The largest known prime number 2^82,589,933 − 1 is a Mersenne prime found on December 7, 2018 with 24,862,048 digits 

2^13 - 1 = 8,191    2^17 - 1 = 131,071    2^19 - 1 = 524,287, BUT 2^23 - 1 = 8,388,607 = 47 x 178,481

5. Listing the numbers in just six columns we see that all primes except for 2 and 3 are found in just two columns. I had the children test this theory on page 1 of the attached pdf.

6.   Leonhard Euler, came close to finding a function of primes, 40 in a row until: 

41 + 2 = 43, which is prime.

43 + 4 = 47, which is prime.

47 + 6 = 53, which is prime.

53 + 8 = 61, which is prime.

61 + 10 = 71, which is prime. 

71 + 12 = 83, which is prime. 

83 + 14 = 97, which is prime. 

97 + 16 = 113, which is prime. 

113 + 18 = 131, which is prime. 

131 + 20 = 151, which is prime. 

151 + 22 = 173, which is prime. 

173 + 24 = 197, which is prime. 

197 + 26 = 223, which is prime.

223 + 28 = 251, which is prime. 

251 + 30 = 281, which is prime.

281 + 32 = 313, which is prime.

313 + 34 = 347, which is prime.

347 + 36 = 383, which is prime.

383 + 38 = 421, which is prime.

421 + 40 = 461, which is prime.

4th and 5th grade: Euler’s pattern can be written as a function where you can substitute whole numbers 1-39 for n so: f(n) = n^2 + n + 41 is prime; then at n=40 it generates 1681 = 41 x 41 and n=41 generates 1,763 = 41 x 43—TOO BAD EULER. The students can find each Euler prime on page 2 of the pdf.

I challenged students to explore patterns of prime numbers like I have for the past 51 years. The fact that my patterns always seem to hit a roadblock has just inspired me more to try to solve this mystery. The Fields Medal and cash prizes are less inspiring for me but can be very motivating for some mathematicians. Let's find this solution together.