A plus sign accompanies the primes; a minus sign, the non-primes. The infinite product of fractions can be inverted because the product equals 1. Equation 36 represents a novel way of defining the number 1. It can also be written as follows:. Equation 36 is obtained as follows. One takes as point of departure the value of L. One next inverts Equation 37 and multiplies the inversion of 37 by Equation 35 above. This means that the fractions pertaining to the non-primes in 35 survive as they are. By contrast, each fraction pertaining to a prime is multiplied by a second fraction pertaining to the same prime.
In the case of the prime 2, the resulting product is as follows:. And transferring to the other side of the Equation yields Equation As was noted, inversion of There are countless related infinite products. But there are also countless related infinite sums. The following example has already been cited at the end of Section This product concerns the primes. It can also be written in a form that resembles 37 more closely, namely as follows:.
The search for the solution of the infinite sum in 39 is known as the Basel problem. As was already noted, L. Euler solved it. Similar behavior can be seen in the following infinite sum involving the uneven numbers, discovered independently by both G. Leibniz and J. Gregory  :. According to the material presented in the preceding sections, there seems to be nothing special about the prime sequence in its relation to other number sequences. Then again, there are opportunities to observe the possibility of prime primacy, so to speak.
It is not entirely clear what to make of it. In any event, any hints at the primacy of primes also are subject to the observations already made above. The primacy involves the factor of infinity in a way that makes its comprehension inaccessible to human cognition. Results for the non-primes such as 18 and 38 were derived from the behavior of the primes and not vice versa. Does this point to a certain primacy of the primes over the non-primes? The primes and the non-primes appear to exhibit a balanced relation. Still, the behavior of the non-primes in 35 has been derived from the behavior of the primes see above.
So it perhaps comes as less of a surprise that, if one makes all the fractions explicit and eliminates all the terms common to numerator and denominator, one obtains a different result that exclusively involves the primes. The non-primes in the denominator are all adjacent to primes. Since a non-prime between twin primes—that is, primes separated by 2—are adjacent to two primes, they receive the exponent 4 and not the exponent 2.
What about the special behavior of 2 and 3? It is clear that, 1 , 3 is adjacent to only one non-prime and not to two non-primes like the primes higher than 3, and that, 2 , 2 is adjacent to no non-prime if, like Euclid, one excludes 1 as a number. This structural principle seems reflected in the following equivalent of 42 :. The behavior of the square of 3 somehow seems indicative of the fact that it is bordered on only one side by a non-prime, the plus side. Primes bordered on two sides exhibit the exponent 4 in It therefore somehow seems natural that 3 exhibits the exponent 2, or half of 4.
Then again, the behavior of 3 can be made to look exactly like that of the higher primes by means of the following suggestive equivalent of 42 :. And in fact, in the following equivalent of the left-hand side of 42 , every single prime can be seen behaving in the same way, accompanied in the numerator by the number that is one smaller and the number that is one larger:. This Equation is all about primes in the denominator and numbers that are one more or one less than a prime in the numerator.
This Equation is all about the primes in the denominator and numbers that are one more or one less than a prime in the numerator, but excluding 1 as a number with Euclid. This Equation is all about the non-primes in the denominator and numbers that are one more or one less than a non-prime in the numerator. It is a bit as if squaring the non-primes yields the primes. One is once more reminded of being in the presence of a number harmony.
But again, the harmony takes place in the dimension of infinity. The deeper nature of the harmony is therefore not accessible to rational human intelligence. The whole matter just again and again keeps evading and eluding rational human intelligence. But who would dare to state that rational human intelligence knows no boundaries and is as large as the universe itself? Another case in which a certain prime primacy may possibly be discerned is as follows. Again, if there is indeed prime primacy, then it is at the same time obvious that the organized behavior is not accessible to human cognition.
The first are as follows:. Since B. Riemann expanded L. Higher values can be found on the Internet . It is also possible to establish values for an infinite prime product that differs from the one above only in that one finds plus where the product above has minus. It may be called the infinite prime product-plus. The first values are as follows:. I submitted the sequence of the integers in the numerators and the sequence of the integers in the denominators of this infinite prime product-plus to the Online Encyclopedia of Integer Sequences www.
The website administrators have computed and added values for higher exponents. Another infinite series is also related to L. It may be called the infinite prime product-minus-plus. It is as follows:. It was in order to submit these sequences that I originally for the first time consulted www. A search quickly revealed that B. Cloitre had already submitted them in February of They are sequences A and A The proof provided below indicates that I reached the same results independently.
No explicit proof is otherwise provided for A and A on www. Therefore, I do not know at this time whether there is another way of providing proof. Cloitre notes, as a remarkable property of the results of the infinite prime product-minus-plus, that they are rational. The reason for this property is explained below. The infinite prime product-plus and the infinite prime product-minus-plus are obtained as follows. Dividing 45 by 46 obviously, after simplification, yields a manifestation of the desired infinite prime product-plus, namely.
But on what does 47 converge? In order to obtain 47 , 45 was divided by It so happens that it is known on what 45 and 46 converge. All that one needs to do, therefore, is to divide the number on which 45 converges by the number on which 46 converges. It is well-known and has already been noted above that. Evidently, dividing 45 , or , by 46 , or , yields. The division in question looks as follows:. Since 47 is itself the result of dividing 45 by 46 , dividing 47 by 46 is in effect the same as dividing 45 by the square of The procedure can be presented as follows:.
Other values for have already been listed above. What is left is a rational number. But it is not clear whether the results for exponents higher than 2 are similarly elegant, probably not. The procedure to establish 18 for exponent 2 in Section Expression 49 therefore is the same as. The left side of Equation 50 is an infinite product beginning at and increasing. The right side is an infi-.
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It is obvious that both infinite products converge on the same number from two directions. They meet at roughly 1. It can be concluded that certain types of results that can be obtained for primes cannot be obtained for non- primes. Is this an indication of prime primacy? If it is, the main point of the present article is once again illustrated, namely that there is abundant evidence of organization in number sequences such as the prime sequence; but also that all this patent organization is not accessible to human cognition.
Aristotle writes as follows about axioms in his Metaphysics , at 4. A single thing cannot at the same time possess and not possess the same attribute all being the same. Some ask for proof , but only because they lack education. Proving everything is definitely impossible. One would just recede into infinity [that is, in trying to prove everything by something else ]. My translation. Distinctions are the stuff of which knowledge is made. Aristotle makes three distinct statements about axioms. First, more specifically, he describes an axiom, the one that he deems to be the most fundamental one of all.
It is also known as the fundamental axiom, or law, of thought. After Aristotle, it became common to think of the existence of three fundamental axioms of thought. The second, more general, statement that Aristotle makes in the passage quoted above is that there must be such a thing as axioms. Nothing seems more evident. A proof of a theorem requires given and accepted observations that are outside that theorem. But if the accepted observation is itself always a theorem, there could be no end to the proof. Each time that one stops at a certain theorem, one would not be finished because the theorem in question would also need to be proven by yet another theorem.
Without axioms, proofs would need to run on into infinity, which is impossible. But then there is no proof for C and E. C or E already prove A, B, or D. However many arrangements one comes up with, at least one theorem ends up without a proof. The same may be assumed if one could consider all the theorems of mathematics all at once. There is no reason why things would be any different from what one finds in the above simple scenario. Is there more to the Incompleteness Theorem than what Aristotle stated already more than two millennia ago? The third statement concerns the order of the axioms.
In that regard, the fundamental axiom of thought stands at the very beginning. It does not even depend on another axiom. What does this mean for the prime sequence? If something outside the sequence explains it, then that would still need to be explained by something else. Or perhaps, many tacitly assume and hope that the deepest structure of the Book of Nature would be magnificently revealed if the prime sequence can be explained, for example, by proving the Riemann hypothesis.
But I believe that this is not possible. The brain is a finite material tool. It can do certain things. But it is certain that there are things that it cannot do. Therefore, there is a line between the two. The thesis of the present paper is that nothing can explain the prime sequence. But even if one assumes for the sake of the argument that something could, then what explains it would still require an even deeper explanation. In other words, there are no final explanations of nature accessible to the limited physical brain in its current state.
This approach makes it also easy for the physical brain to be completely comfortable inside its absolute limitations once it realizes that there simply must be limitations to something that is physically finite. This approach makes what are considered abstruse mathematical concepts delightfully simple. Consider infinity. My own view is that nothing is easier to understand than infinity. Additional remarks on infinity and the pesky concept of the infinitesimally small follow in Section What about infinity? The matter is simply this. However great a certain number may be, one can always add 1 and make it bigger.
And one can just keep doing this. There is no reason to stop.
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What could be more obvious? That is clear, as has already been pointed out above. What is easier to accept than that 17 is a prime? An axiom is an observation that is so abundantly obvious that no one sees a need for a proof. One simply accepts it without proof. Everyone agrees without proof that, say, 7 and 11 are primes. I personally know of no one who has ever asked for a proof that 7 and 11 are primes. That makes their identity an axiom. And that is also how Euclid appears to treat their identity see below.
It is true that, in terms of identity, it becomes ever more difficult to identify ever higher primes.
Does this make the prime sequence less than abundantly obvious and therefore less of an axiom? The difficulty with finding ever higher primes has to do with limitations of human intelligence and, beyond that, of computing power. If one had all the time in the world if not more and boundless intelligence, one could establish very high primes by the very same principles by which one establishes that 7 and 11 are primes.
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Physicists have long stopped looking for final causes and perhaps mathematicians should likewise become a little less ambitious. Lagrange writes, as translated from the French  :. The purpose of this work is to reduce mechanics to purely algebraic operations. Maxwell did not know what an electron is. In sum, I am at peace with the notion that the prime sequence exhibits a perfect structure that I will never be able to understand.
Much of what has been written above involves infinite series. Infinite series expose much—one might even say, an infinite amount of—organized behavior on the part of the primes, the non-primes, the even numbers, and the uneven numbers, and so on. But it reveals this behavior only in the dimension of infinity, which is inaccessible to human cognition.
Infinite series have played an enormous role in the development of mathematics since early modern times. They have revealed many truths. But the series also imply by their very nature that these truths are located in the dimension of infinity. The concept of partitions has already been introduced and briefly described in Section 11 above. It is my impression that the organization of the primes should have something to do with partition. Yet, I have so far not been able to find any reference to partition in connection with the prime sequence in standard descriptions of number theory including those by G.
Andrews, T. Apostol, U. Dudley, G. Wright, E. Lucas, O. Ore, and others. For example, in the chapter on partitions in G. Then again, the literature on number theory is enormous. The reason for the absence may well be that, while the concept of partitions must be relevant to the prime sequence, it appears upon closer inspection that there is little or nothing that partition theory can say about the prime sequence at this stage. And it seems probable that partition theory never will, even if primes have everything to do with partition.
In that regard, the concept of partitions supports the notion maintained in the present article that the organization of the primes is ultimately not accessible to human cognition. What are partitions? Partitions are a property of numbers. They are the ways in which an integer can be equated to, or partitioned into, sums of integers. Partitions are therefore in effect sums and might be called partition sums. Partition s theory is the branch of mathematics that studies partitions.
An easily accessible introduction to partitions by J. Tanton, who has announced a book on the subject, is available on the Internet . I adopt a number of general observations from his account in what follows. For example, there are four ways of partitioning the integer 4 if order is not considered important.
The resulting partitions or partition sums are as follows:. Partition theory is not just about identifying all the specific partition sums. Partition theory is concerned mainly with ways of determining by means of general formulas how many possible partitions there are. That does not only involve determining how many partition sums there are in general but also how many there are of a certain type.
The distinction again applies between disregarding order or taking it into consideration. The problem of partition theory that attracts perhaps the most attention—not in the least because it has not been solved—is determining how many partitions there are in general of a given integer n if order is not important. The desired number is signified by the expression. Thus, in the example mentioned above,. Ramanujan first guessed, and then later proved together with G. Hardy, that is approximately.
An example of a type of question for which partition theory does offer a solution is as follows  : In how many ways can 12 be partitioned into sums of three sum components? There are 12, as follows:. This question calls to mind indeterminate algebra and Diophantine Equations. But this is not the place to elaborate on the matter.
Another question that has been answered in general and involves types of partitions is as follows: How many partition sums contain all even numbers or how many contain all uneven numbers? The number of integers is infinite. No precise numbers can therefore serve as the answers to these questions. Instead, L. Euler proved that there are as many partitition sums involving all even numbers as there are partition sums involving all uneven numbers, without however providing a formula to determine how many exactly there are of both.
Interestingly, L. Euler used infinite series in his proof. The present paper proposes that no formula can be found for the prime sequence because the organization of the primes plays out in the dimension of infinity, which is not accessible to human cognition. Primes may well be the subject of one such question.
Primes are integers that have no partition sums consisting of two or more equally large sum components, excluding 1 as a possible sum component. More on this below. The question that partition theory might ask about primes is therefore as follows: Can it be determined on the basis of one formula whether a certain integer can or cannot be partitioned into equally large sum components? Those that cannot be would be the primes. Partition theory seems like a vast area of mathematics in which so much still needs to be discovered. Or perhaps it is a much smaller area than may be assumed because so much will forever remain opaque.
It is all about dividing numbers into types of sums. However, beyond number theory, an integer can refer to a number of things in reality. These things can all differ from one another. What is more, these things can come in different orders or arrangements. And the question may be asked in how many ways a certain number of things can be arranged. In this case, the number of partitions can be determined exactly. Combinatorics plays a role in determining the numbers. Combinatorics is the field of mathematics that teaches one in how many ways 12 people can sit on 12 chairs.
The answer is 12! The first person can sit on 12 different chairs. In each of these 12 different ways of sitting, 11 chairs are left and the second person can therefore sit on 11 different chairs. In other words, for each of the 12 ways in which the first person can sit, the second person can sit in 11 ways.
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And so on for the third person and all the other persons. It is easy to see the difference between partitions and the problem just discussed. A question involving combinatorics that does exhibit similarity with the partition theory component of number theory is as follows: In how many ways can one hang 20 flags on 12 poles to broadcast different signals?
The assumption is that there may be poles without flags. It is with the flags hanging on the poles as with the persons seated on the chairs above. The way of arranging 20 flags in a sequence is 20!. But there is more to be arranged than just the flags. The sequence of the flags may be disrupted by a shift from one pole to the next.
So the beginning of a sequence may be as follows: flag, flag, pole shift, flag, pole shift, and so on. That means that there are 2 flags on the first pole and 1 flag on the second pole. The beginning of another sequence is as follows: flag, flag, flag, pole shift, flag, flag, flag, pole shift, and so on.
That means that there are 3 flags on the first pole and 3 flags on the second pole. Between 12 poles, there are 11 shifts of poles. It appears that 20 flags and 11 pole shifts need to be arranged, or 31 items. If the 31 items were all different, as is the case with the persons seated on the chairs above, the number of the arrangements would be 31!
But whereas the 20 flags are all different, the 11 pole shifts are all the same. It is therefore necessary to divide 31! It was noted above that. For let the reference be to four flags. How does the classical partition problem exemplified above by —or that 4 flags can be partitioned in 5 ways—differ from the problem involving 20 flags and 12 poles? One difference is that arrangement of the flags, the sequence in which they come, plays no role in the classical partition problem whereas it does in the problem involving 20 flags and 12 poles.
By arrangement, I do not mean order of the quantities of the flags, in that arranging 5 flags on a first pole and 10 on a second is different from arranging 10 on the first and 5 on the second. To get closer to the classical partition problem, arrangement may be made irrelevant. This means that the difference between placing flag a above flag b on a pole and placing flag a below flag b on the same pole does not make for a different signal.
How so? Each of the 20 flags can hang on each of the 12 poles. If there were just 2 flags, the first could hang on each of the 12 poles and so could the second. Evidently, one needs to multiply 12, the number of positions of one flag, as often with itself as the number of the flags, that is, The total number of the combinations is therefore 12 The 20 flags are still all different, but not in. To get even closer to the classical partition problem, let the 20 flags be all the same.
How close is this? Close but not the same. This partition into 20 portions is not possible because there are only 12 poles. What can be solved at this time, apparently, is the number of partitions of 20 flags into 12 portions. But the need is for the number of partitions in 12 portions or less. As far as I know, there is no solution to this problem. By order, I evidently do not mean order in the sense of arrangement of the flags.
After all, it is not possible to place the flags in different arrangements in the second problem of flags and poles because they are all identical. Much more could be said about partition. The afore-going remarks are meant to provide general background. Partition theory as a branch of number theory was a preferred subject of study of the celebrated, and slightly enigmatic, mathematician S. Ramanujan Before him, L. I am not sure whether S. Ramanujan ever associated partition with primes or was ever inspired to study partition because of primes.
I have not engaged in any in-depth investigations. Advances keep being made in the theory of partition, even in most recent time. New results obtained by J. Bruin and K. Ono were recently widely publicized. Partition is not an easy subject, especially when order is not important. The subject swiftly moves into higher mathematics as soon as—beyond the basic definition—past results need to be discussed when order is not relevant.
Thus, in a page introduction to number theory, partition is only mentioned briefly in a historical account of S. Mathematics is ultimately a reflection of how the brain perceives reality. It is customary to define primes and non-primes in terms of their factors. The focus is on multiplication. But as has been already noted in Section 11, multiplication is just an abbreviation of addition, as follows:. It is in regard to addition that there is a more direct way in which primes and non-primes make a statement about reality and directly affect certain very practical aspects of daily life.
Consider a number of people needing to divide into equal groups. A prime number of people cannot divide into a number of all equal large groups. A non-prime or composite number of people can. The need to divide something into equal groups may be of concern in all kinds of practical situations. What if camp leaders at a summer camp are faced with 31 boys and need to divide them into equally large groups for all kind of tasks?
If a soccer tournament were organized involving all the boys, it would not be possible to field teams that are all of equal size. The same with cleaning crews. And so on. It is evident that a prime has partition sums just as much as a non-prime. Only, the sum components of a prime are never all the same. What kind of answers might the student of the primes and the prime sequence want to expect from partition theory? The need is for a single function or a single formula or the like that could establish whether or not the integers of each of the partition sums of a certain integer are never all equal.
There is a procedure for establishing in how many partition sums of an integer all integers are unequal . This result is somewhat close to the partition of primes into unequal sum components. But it is not nearly close enough. It is removed from the primes by at least three steps.
The first step of removal is that the procedure does not concern all partition sums of a certain integer. It singles out only those that exhibit a certain property, in this case all those whose integers are all different. The need is for a procedure allowing a statement about all partition sums of an integer. The second step of removal is that the property of the partition sums in question does not cover all partition sums pertaining to primes.
Take, for example, the prime 7. The partition sums. The procedure at hand only captures partition sums in which all integers are different. The procedure cannot, therefore, isolate the primes. The third step of removal is as follows. Let us assume for the sake of the argument that there indeed exists— though there apparently does not—a procedure that identifies the exact number of the partition sums whose integers are not all equal, even if the integers of some other sums may be equal. This procedure would not exclude the possibility that, in some partition sums, all integers are indeed equal.
How to establish that, in none of them, the partition sums relating to a certain integer contain integers that are all equal? Establishing this would make it possible to identify the integer in question as a prime. There exists a procedure discovered by L. Euler that generates all the ways in which any number can be partitioned, with order not being taken into consideration .
Bruiner and K. Ono have recently discovered a finite formula for computing partition numbers. Rademacher had earlier discovered an exact convergent series for the partition function p n , improving upon S. Accordingly, if one would compare the number of all the possible partition sums with the number of the partition sums whose integers are not all equal produced by the fictional procedure mentioned above, it would be possible to establish whether the integer that is being partitioned is a prime or not.
In the case of a prime, the two numbers would be the same. In other words, there would be no partition sums whose integers are sometimes all equal. That would be a fine result. But the key point is this. This fictional manner of identifying primes would still only identify the primes a little bit in the way that the Sieve of Eratosthenes does. It is not clear whether it would say anything about the prime sequence itself. Show More Show Less. Add to Cart. Any Condition Any Condition. No ratings or reviews yet. Be the first to write a review. You may also like. Gerald Durrell Paperback Books.
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Atkinson, H. Handley, D. Holton and D. This is all encouraging. Maybe more feasible would be to have some distributed system of open databases, and there are some exemplars out there, in very structured environments. There's potential for doing the same sort of thing with theorems Alekseyev, Josephus problem , The Empire of Mathematics, 2, Alekseyev, On the number of permutations with bounded runs length, Arxiv preprint arXiv : Alekseyev, Computing the number of inverses of Euler's totient and other multiplicative functions, arXiv preprint arXiv : CO] Max A.
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