Here's an "oldie but goodie" for you this time!Some time long ago a mathematician, named C. Goldbach, was playing around with prime numbers and noted the following oddity: This is really a simple idea, as these few examples will show: 20 = 7 + 13 34 = 3 + 31 42 = 13 + 29 66 = 19 + 47 72 = 11 + 61 100 = 47 + 53 So what's so hard about that, you ask? Well, just try proving that it is true for ALL even whole numbers! There's the rub. Ever since Goldbach made this observation, mathematicians, professional and amateur alike, have tried to do that. But with no real success. And that's why it is called a "conjecture", my friends. A conjecture is a statement that you think is probably true, at least based on all the information at hand. But a "for sure" proof is not available. You see, it is not enough, as in a case like the one before us, to just say, "See! All the even numbers I've tested so far can all be expressed as the sum of a pair of primes. So all of them must behave the same way." Mathematics doesn't work that way. ## All even numbers, greater than 2, can be expressed as the sum of two primes.

So where do we go from here, you ask? Well, one of my favorite activities for my students is to ask the question: This can be shown quickly by an example: 24. 5 + 19 7 + 17 11 + 13 So 24 can be expressed with ## How many pairs of primes can be found for any specific given even number?

pairs of primes. For those interested in making this investigation a bit more elegant, we can define this with function notation of advanced algebra in the following way:threeSo for our example, we would write ## Let GN(

n), the "Goldbach Number ofn", be defined as the number of pairs of primes the sums of which are the numbern.GN(24) = 3 .Now, it's easy to see that this is a good function, as every n(input) has a single GN(n) (output). But what is not so obvious is this fact: as the value ofnincreases, the same can not always be said for GN(n). Here's a specific example of what I am saying here: 24 < 26 but GN(24) = GN(26) because GN(26) also equals 3. [3+23, 7+19, and 13+13] And even more significant is the case for n = 26 and n = 28. Observe: 26 < 28 but GN(26) > GN(28) because GN(28) is only 2. [5+23 and 11+17] And there are many more such instances of this phenomenon. How many can you find? Speaking in the language of algebra, we could say:Now here's your assignment, class... ## If n < n+k for some positive integer k, it is not always true that GN(n) < GN(n+k).

Find the GN values for all the even numbers from 4 to 100.To assist you with your work here, a nice table of all the primes between 1 and 1000 can be found by clicking HERE.## Extension

Why stop with pairs of primes and the even numbers? Let's also usetriosof primes and go for theoddnumbers. Mr. Goldbach did, or so I hear. You surely will find plenty to challenge you here.

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