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The concept of prime number is one that has fascinated mathematicians and frustrated students for a long time. It is one of the most fertile fields where one can make interesting discoveries on any level one wishes. By way of illustrating this, you only have to check out a few of the pages in this website: “Sexy” Primes, Primes & Square Pairs, Big Primes, and Paul Erdös. And a few more are in the planning stage.
I wish to turn my attention to something that is not so well known in the literature, but based on a math problem I once saw in a book. It concerned the sum of 3 primes that was itself a prime. The details are lost in my memory as to the exact numbers used, except to say that the problem, as stated, had an error in it. But it set me to thinking about the idea that will be explained here.
We begin in a time-honored fashion, with a couple of definitions:
Consecutive-Prime Sum (CPS): a number that is the sum
of two or more consecutive prime numbers.
Prime-Sum Prime (PSP): a prime number that is the sum
of two or more prime numbers.
A moment’s reflection on these definitions is necessary before proceeding further. A CPS is nothing more than a number that results from chosing any number of consecutive primes and finding their sum. That sum may take on various characteristics: prime or composite, naturally, or such things as squares, palindromes, etc. Whereas a PSP results from choosing any prime numbers one wishes (consecutive or not, with repetitions, etc.); it’s just that the sum itself must be prime.
Now, if we combine these two ideas into one, we really have something special:
Before we get lost in this alphabet soup, let’s look at a few examples:
|CPS:||13 + 17 + 19 = 49|
|PSP:||11 + 13 + 19 = 43|
|CPSP:||11 + 13 + 17 = 41|
Do you get it now? In the first case, the primes were consecutive; yet the sum did not happen to be a prime, in fact, it was a square number. In the second case, the primes were not consecutive, but the sum was a prime. But in the last case, the primes were consecutive AND the sum was also a prime. There we get the “best of both worlds”, as it were.
Well, now that that’s out of the way, where do we go from here? We have several options actually. One is to determine how many of the CPS‘s from 10 to 100 are themselves prime, that is, CPSP’s of order-3. In other words, limiting ourselves initially to three consecutive primes, how many solutions are there to this equation?
Pn + Pn+1 + Pn+2 = Pk
As it turns out, there are seven values for Pk, one of which was given above (41). We leave it to you to find the remaining six. Next, you can try four consecutive primes. Now you are looking for CPSP’s of order-4. [You should shortly make a very fundamental discovery when working on this case.] Later, you should pass on to 5 consecutive primes, etc.
A piece of advice is perhaps in order here: this now begins to be a good situation where you could use a spreadsheet to make the work more efficient.
There you have it! Now the rest is up to you. Dive in and see what you can discover.
To make all this prime hunting a little easier, we provide you with a chart of all the primes between 1 and 1000.