a search of primes from lesser primes

A search for primes from lesser primes

Jean Dayantis ∗

12, rue de l’Olivette34570 PignanFrance

Jean-Francois Palierne

Laboratoire de PhysiqueUMR 5672, ENS-Lyon46 allee d’ltalic69364 Lyon Cedex 07France

Abstract

The purpose of this paper is to show that there is an algorithm which permits to definenew primes from lesser already known primes. The method is based on the definition of“compact” sets of primes, and on the fundamental property, i.e., the fact that the sum ordifference of two mutually prime integers have no common decomposition factors with thesetwo mutually prime integers.

Keywords and phrases : Integer and prime numbers, search for primes, Eratosthenes sieve,

diophantine equations, primorial function.

Introduction

We want here to present some properties of prime and compoundinteger numbers which possibly may have not so far attracted the attentionof number theorists. Insofar as the properties here investigated presentsome interest, the authors have found no reference to the subject hereaddressed in several books on number theory [1]-[7]. It is of coursequite possible that such references could be found within the numerous

∗E-mail: [email protected]—————————————————–Journal of Discrete Mathematical Sciences & CryptographyVol. 10 (2007), No. 4, pp. 581–602c© Taru Publications

582 J. DAYANTIS AND J. F. PALIERNE

publications which have appeared on number theory in the past. Even so,however, it is hoped, as the result of the very rapid recent progresses incomputational means and performances, that the present paper may addsome new and interesting results to the topic.

1. General principles

1.1 Decomposition of an odd integer in a binary sum or difference of products ofprimes

Let A and B be two mutually prime integers, which factorize inproducts of primes:

A = a1a2 . . . am (1)

B = b1b2 . . . bn (2)

such that no ai is equal to a b j . In relationship (1), it will be assumed thatfor example one may have ai = ai+1 or further ai = ai+2 , and so on.This will provide for the possibility that ai appears in fact as some integerpower larger than one in the factorization in primes of A. The same remarkapplies of course for relationship (2). Let us now consider the sum

C = A + B . (3)

According to our assumption, both A and B cannot be even, and it will beassumed that one of the two components in the sum is odd. It may nowbe shown that C factorizes in a product of primes C = c1c2 . . . cp where nock can be equal to any ai or b j ; for, assume for instance that ck = b j . Then,the sum A + B− C = 0 could have been written in the form

A + B− C = a1a2 . . . am + b j(b1 . . . b j−1b j+1 . . .

bn − c1 . . . ck−1ck+1 . . . cp)

= a1a2 . . . am + b jd1d2 . . . dq = 0 , (4)

where b jd1d2 . . . dq represents the factorization in a product of primesof B − C. However, relationship (4) is impossible, due to the uniquedecomposition of an integer in a product of primes within the field Z ofalgebraic integers [8], and to the fact that by hypothesis b j cannot be foundamong the primes ai . The same argument may be repeated assuming thatA and C share one or more primes, so that the conclusion follows that theai ’s, b j ’s and ck ’s are all distinct primes.

A SEARCH FOR PRIMES 583

This property is of importance for present purposes, and will becalled the fundamental property of a binary sum, where the two elementsA, B, in the sum share no common primes. Relationship (2) can begeneralized by writing

C = A− B . (5)

Of course the fundamental property applies also in this case. As will beseen later, it is more important to consider differences A− B rather thansums A + B.

1.2 The fundamental property does not generalize to a sum or difference of morethan two products of primes

Let us for example write

C = A + B + E , (6)

where A, B, and E are three pairwise relatively prime integers (nocommon divisor between A and B, A and E and B and E respectively). Ifthen one assumes, for example, that ck = b j , one can write

A + E + (B− C) = a1a2 . . . am + e1e2 . . . er − b jd1d2 . . . dq = 0, (7)

where b jd1d2 . . . dq stands for the factorization in primes of B− C. Now,because of equation (6), the factorizations of A + E and C− B are identical,and eq. (7) is always possible, even though B and C share commonfactorization primes. C may now share common divisors with A, B andE, though this is not necessarily so. Some numerical examples will makethings clear:

1. Take A = 2 × 3 = 6, B = 5 and E = 7, so that C = 18, whichshares common divisors with A.

2. Take A = 2, B = 3 and E = 5, then C = 10 = 2 × 5, so that Cshares common divisors with A and E.

3. Take now A = 4, B = 5, and E = 3× 7 = 21, so that C = 30 hascommon divisors with all three elements in the sum.

4. Finally, take A = 7, B = 11, and E = 13, so that C = 31. Here wehave four distinct primes, and therefore no common factor. Noticethat is not necessary that A, B, C and E be all primes in order thatC shares no common divisors with A, B, and E.


The conclusion drawn from these numerical examples follows: if oneconsiders the sum C of three mutually prime integers, the factorization ofthis sum may or may not contain common factorization factors with oneor more elements in the sum. This is the essential difference with the caseof the sum of two mutually prime integers considered previously, wherethe sum never has common decomposition factors with the two elementsin the sum.

The same conclusion holds if B or E or both have a minus sign,provided that, for present purposes, C remains a positive integer.

Of course, one may proceed on and address the properties of thesum C of four or more mutually prime integers, but this seems devoidof interest and will not be considered here.

1.3 Compact sets of primes

Consider the ordered set of primes q1 = 2, q2 = 3, q3 = 5, q4 = 7. . .. (We shall put in what follows p0 = 1. Unity is considered here as aspecial prime number, since it is divisible by itself and by 1; it is a kind ofdegenerate prime number). We now say that a set of primes up to prime qs

is compact if it contains all primes up to qs , to the exclusion of any largerprime. The compact set of primes 1, 2, 3, 5, . . . , qs will be denoted by qs.Alternatively, when the last prime in the set is specified, let it be P, we maymore simply denote it by P. Thus, the compact set of primes 1, 2, 3, 5will be denoted either as q3 or 5. (Notations according to the rangeor the numerical value of the prime).

1.4 How many ways to write an arbitrary integer as a binary sum or differenceof products of primes. Permutations of factors in the decomposition of an integer

If the integer C is even and we decompose it in A + B, there areΦ = C/2 such ways of decomposition; if C is odd, then there areΦ = (C− 1)/2 ways of decomposition. For example, for C = 10, one maywrite 10 = 5 + 5, 6 + 4, 7 + 3, 8 + 2, and 9 + 1, in all five ways. However,instead of considering the decomposition C = A + B (eq. (2)), one mayconsider instead the decomposition C = A − B (eq. (5)). Here clearlythere is an infinite number of decompositions, by incrementing A and Bat the same time by one or any arbitrary integer. Let then be the integernumber C = 30 = 1× 2× 3× 5, which is the primorial function of q3.From C one may obtain a set of new integers by permuting the elementsof q3 in a binary sum A + B as follows: C1 = 1 + 2 × 3 × 5 = 31;


C2 = 2 + 3 × 5 = 17; C3 = 3 + 2 × 5 = 13; C4 = 5 + 2 × 3 = 11; thisnew ensemble of integers will be denoted by Q4, 1, the 1 standing forthe fact that the above substitutions take account of only the first powerof the primes in q3. More generally, one may denote by Qs, z theset of substitutions derived if one considers the primes in qs up totheir zth power. An analogous process may be defined if one now writesC = A− B.

1.5 Prime numbers from the set qsTheorem. If a permutation Ck = Ak + Bk or Ck = Ak − Bk in the decom-position of an integer C in a binary sum or binary difference of products of primes,forms a compact set qs of primes, and if qs < Ck < (qs+1)2 , then Ck is prime.

Proof. First, according to the fundamental property of paragraph 1.1, Ck

cannot be a prime among those found in qs, with the exception howeverof unity. (The case of unity is specific and will be discussed in paragraph2.2.) Let us now assume that qs < Ck < (qs+1)2 and that Ck might beexpressed as a product Ck = P1P2 of two prime factors P1 and P2 . SinceCk is less than (qs+1)2 , P1 and P2 cannot both exceed qs+1 . Therefore,only the cases P1 , P2 both less than qs+1 , or P1 equal or greater than qs+1 ,while P2 is less than qs+1 , should be considered. In the former case, bothP1 and P2 are assumed to be primes less than qs+1 ; this however is notpossible, because the integers in A and B form a compact set of primesqs and according to the fundamental property, C = A + B or C = A− Bcannot share primes included in the compact set qs; in the second case,the same may be said for the factor P2 , which will contain factors includedin qs. Thus P2 is at least equal to qs+1 so that the product Ck = P1P2 willexceed (qs+1)2 . Therefore Ck cannot be a product P1P2 and consequentlyis a prime. Let us notice that the proof of this theorem may also be derivedusing Eratosthenes’ sieve method [9].

The above theorem is a special case of a more general theorem, whichstates that if a partition of the elements of the compact set qs in A and B suchas Ck = Ak + Bk or Ck = Ak − Bk , with (qs+1)n < Ck < (qs+1)n+1 , then Ck

is either a prime or a product of at most n primes. (The words partition andpermutation used previously have here essentially the same meaning.) ¤

The proof follows the same lines as above: Ck being larger than(qs+1)n and less than (qs+1)n+1 (or (qs+1)n+1 − 1, the two conditions are


equivalent) can not clearly be decomposable in more than n prime factors,since each prime factor is at least equal to qs+1 . The least possible value ofCk occurs when Ck is the nth power of prime qs+1 . Then, in some cases, Ck

can be the product of n primes close to qs+1 , provided that their productdoes not exceed (qs+1)n+1 . Most often, however, Ck will be the product ofless than n prime factors, each factor displaying its own power; as alreadyindicated, Ck can also be a prime.

To give examples, let us focus our attention on q3 = 5: therange where Ck is a prime is 6 . . . 48 (since 48 = 72 − 1), that whereCk is either a prime P or the product of two primes P1P2 , is 49 . . . 342(since 342 = 73 − 1), that where it is either a prime P or the product ofat most three primes P1P2P3 is 343 . . . 2400, and so on. One checks that2× 3× 5− 1 = 29, 3× 5− 22 = 11 and 22 × 3− 5 = 7 are primes; that22× 32× 5− 1 = 179 is a prime, but that 32× 52− 22 = 221 is the productof two primes, 13× 17. Generally speaking, partitions of 5 outside therange of confidence for primes and up to 75 − 1 = 16806 are in majorityprimes, with many being the product of two primes and a few of threeprimes. No partition was found in the range 343 . . . 2400 which was theproduct of three primes, although allowed by theory. (For more details seeAppendix B).

Corollary. Let a compound integer C be the product of n primes, none of whichis contained in the compact set qs. Then, no partition of the primes in qsmay obtain the compound number C, unless C is greater than (qs+1)n .For, if m < n, values of Ck = Ak − Bk such that (qs+1)m < Ck < (qs+1)m+1

can have, according to the previous theorem, at most m prime factors.

1.6 Reciprocal of the theorem in paragraph 1.5

Problem. The question is now asked whether any prime P such thatqs < P < (qs+1)2 − 1 can be expressed as a sum of products of primesA + B forming a compact set qs. This is the reciprocal of the theoremin paragraph 1.5. The answer is no, since one can easily find primes, inthe range of a few hundreds, which do not satisfy to that condition, thatis cannot be expressed as a sum A + B, where the integers A and B inthe sum contain all the primes of the set qs, and only these primes.This classifies all primes in two distinct categories, those which can beexpressed by a sum A + B satisfying to the ascribed above condition, andthose which cannot.


The same question asked for the difference A − B is much moredifficult to answer, since there is now an infinite number of ways forwriting C = A− B. The remainder of the article will be essentially devotedto try answering that question. For small primes the permutations of theprimes of the corresponding compact set qs within A and B permits toreach all the primes in the range of confidence qs+1 , (qs+1)2 − 1. However,for larger primes, computer numerical investigations show that this isno more the case. One may wish to address the problem through theknown theorem [10] that if A and B are mutually prime integers, thenthe diophantine equation

Ax + By = C (8)

is always soluble, i.e. one can always find integers x and y which satisfythe equation. Putting A = a1a2 . . . am and B = b1b2 . . . bn one maywish to find integers A′ and B′ such that A′ − B′ = C, with the twoconditions A′ = A f (A) and B′ = Bg(B), where f (A) and g(B) arepowers respectively of the ai ’s and the b j ’s. In practice, while it is quiteeasy to find separately convenient A′s and B′s, it is much more difficult tofind for large enough primes couples A′ , B′ , which satisfy simultaneouslythe above two conditions. In this search, computer softwares may beefficient help. Generally speaking, computational work, while unable toprovide universal solutions or proofs for our problem, is quite helpful inorienting the research and helping one to avoid spurious conclusions.

As already stated, it is much more interesting to consider differencesC = A− B rather than sums C = A + B. From here on we shall consideronly differences A− B.

2. Computational work

2.1 Basic and overdimensioned compact sets

Definitions. Let P be an integer obtained from a substitution Ck of qs(see 1.4). If s is the least index in an ensemble qs such that P < (qs+1)2 ,qs will be named the Basic Compact Set of P (BCS). If t > s is such thatone still has qt < P < (qs+1)2 , this set will be called an OverdimensionedCompact Set (OCS). While the BSC is unique, there are many OCS’s,however their number is finite. An OCS contains necessarily more primesthan the BCS. Finally, if one considers a set qh such that qh > (qs+1)2 ,this set will be named an Unsuitable Compact Set (UCS). The importance


of the BCS lies in the fact that it is the least compact set that ensures,according to the theorem in 1.5, that P is prime. This is also the case for allthe OCS’s. An index w < s, such that P > (qw+1)2 does not ensure that Pis prime, although it can be one.

We now proceed to indicate the computational work, done in relationto the problem C = A − B. A first software calculated all values A =P + B, with P prime, starting with B = 1. It stopped calculating whenfinding a B leading, together with the corresponding A, to the basiccompact set (BCS). For some value of B, going up to B = 109 , if no suchset was found, the calculation was stopped. Taking the maximum valueof B equal to 109 , the first failure occurred for q78 = 397, whose BCS is19. Then a second software was devised, which determined all compactsets expressing a given prime P as P = A− B. This second software had acapacity for A and B of up to 264 , which is about 1.8× 1019 . Taking againBmax = 109 , going up to q131 = 739, the BCS of which is 23, otherfailures occurred for the following primes: 419, 433, 487, 547, 557, 563, 599,631, 647, 659, 701 and 733.

Now a third software was devised which took advantage of eq. (8)and the remark that follows the equation: this software, instead of tryingfor success all the values of B from one to some maximum value, triedonly the values of the form B′ = B f (B), where B took some initialvalue. This third software saved considerable computational time tofind the decompositions of any prime P in differences P = A − B. Anyinteger C, prime or not, if not too large, may be expressed in numerousways through differences A− B forming compact sets qs, where s cantake many different values. For example, let us take C = 29, which isprime. Computer analysis using the second software showed that thetotal number of compact sets able to express prime 29 up to B = 106 was24. Among these sets only one was less than the basic compact set BCS,twenty were OCS’s, and three were equal to the BCS, which means thatthere are three different ways for expressing 29 using the basic compactset 1, 2, 3, 5. We here give for example four of these sets:

29 = 30− 1 = 2× 3× 5− 1 set q3 = 529 = 32− 3 = 25 − 3 set q2 = 329 = 35− 6 = 5× 7− 2× 3 set q4 = 729 = 260− 231 = 22 × 5× 13− 3× 7× 11 set q6 = 13


All these sets except q2 ensure that 29 is a prime; this is not the case forq2, because its range of confidence runs for integers larger than 3 butless than 52 = 25. For qs equal or larger than P, as a consequence of thefundamental property, P should appear in the r.h.s. of (5) and this in bothA and B, but not necessarily at the same power. This happens in particularif one multiplies by P both sides of an identity

1 = a1a2 . . . am − b1b2 . . . bn . (9)

Thus, it is of limited interest to search for a decomposition of the primeP as a difference of products A− B forming compact sets equal or largerthan P (that is UCS’s), with the exception however of unity (see below).For example, one may write for prime 17 using the UCS q7 (since 17 isthe seventh prime):

17 = 210392− 210375

= 23 × 7× 13× 172 − 32 × 53 × 11× 17 . (10)

2.2 Unity as represented by compact sets

Dividing now both members of identity (10) by 17, one finds:

1 = 23 × 7× 13× 17− 32 × 53 × 11 = 12376− 12375. (11)

This is a representation of unity through the compact set q7 = 17.One also has, using now set qs = 19:

17 = 2× 7× 11× 17× 19− 32 × 52 × 13× 17

= 49742− 49725 . (12)

Dividing again both members of the above identity by 17, one now finds:

1 = 2× 7× 11× 19− 32 × 52 × 13 = 2926− 2925 . (13)

In this instance, the r.h.s. of the identity does not form a compact set(since 17 is absent) and thus is not a convenient representation of unity,according to the philosophy of the present paper. For, any difference oftwo succeeding integers equals unity, but only those differences formingcompact sets enter within the frame of the present work. Since unity hasneither a BCS nor OCS’s, all representations of unity through compactsets are UCS’s. The interesting point is that since unity is inoperativewith respect to multiplication or division, all proper representations ofunity through compact sets may be considered to contain it in bothA and B. This “phantom” presence of unity in A and B ensures that


unity complies with the general property stated above, i.e., that anyrepresentation of a prime P through a compact set qs, with qs > P (thatis any UCS), must contain P in both A and B. The expressions of unitythrough compact sets up to B = 1010 are given in Appendix A.

2.3 Primes from OCS’s

Above in paragraph 2.1, it was stated that 397 was the first primewhich could not be expressed through its BCS when B was restricted to109 . However, program number two showed that 397 could be expressedthrough the next compact set, q9 = 23 as

397 = 8008462− 8008065

= 2× 72 × 11× 17× 19× 23− 36 × 5× 133.

Other failures recorded in 2.1 can also be expressed through OCS’s asfollows:

419 = 12128480− 12128061

= 25 × 5× 73 × 13× 17− 3× 11× 19× 23× 292 ,

557 = 482885− 482328

= 5× 13× 17× 19× 23− 23 × 33 × 7× 11× 29 .

Of course the failure of 397 and other primes to be represented bytheir BCS’s, for B restricted to 109 , but only OCS’s, is no proof that theseprimes could not be represented by their BCS’s for higher values of B.However, this prompts one to generalize the question asked in 1.6 asfollows: Can any prime P be expressed by its BCS or by one or more of itsOCS’s?

For the time being no rigorous mathematical treatment is availablein order to definitely answer the above question. However, numericalevidence strongly suggests, as will be shown in part 3 below, that theanswer is no: there are integers (not only primes) which cannot beexpressed as a difference of products of primes forming a compact set.

3. Tables and figures

It is now convenient to look for more information on how theproportion of successes for representation by compact sets varies whenboth P (P prime) and the compact set qs considered are varied. Statedslightly differently, what the probability is that an arbitrary prime P can


be represented by the compact set qs, (qs being either a BCS oran OCS), as a function of the numerical range in which P lies and thevalue of s? To this end the third software indicated in 2.1 was extensivelyused, the most efficient one with respect to computational time. Table 1displays the proportion of “hits” or successes S per prime as a functionof the numerical range X0 , X0 + ∆X investigated and the compact setconsidered. ∆X was so chosen as to include about 130 primes lyingin the range. This was a compromise between acceptable if not reallysatisfactory statistics and computational time. The computer calculationswent up to the twentieth power of the primes in the compact set, this figurebeing doubled for 2 and 3. In other words, the ensemble of substitutionsinvestigated was larger than Qs, 20 (see 1.4), since primes 2 and 3 weretested up to fourtieth power. Values of A and B exceeding 9.223 × 1018

were discarded, because of the limitations of the computer.For the smaller compact sets, statistics were good enough to show

that the plots of S versus ln X , derived from the data in Table 1, roughlyresemble a branch of a gaussian curve. For the larger sets, due to the smallnumber of successes, the statistics were uncertain. Therefore, no particularattention should be given to the fact that the data for the larger compactsets first increase with X0 before decreasing, or else are very irregular.

It was now admitted that all curves derived from the data in Table 1have the same general analytic form, with varying parameters, and theroughly gaussian appearance of these curves for the lesser sets, not hereshown, prompted us, in the hope of obtaining linear regressions, to plotln S versus [ln ln X]b , the exponent b being an adjustable parameter to bedetermined from the computer numerical data. To do this, the data inTable 1 were considerably extended by taking smaller numerical intervalsX0 , X0 + ∆X , counting the number of successes S′ within this interval,dividing the result by the number of primes within the interval to obtainS, the successes per prime, and then plotting ln S versus [ln ln X]b , whereX was taken equal to X0 + ∆X/2. The parameter b was then adjustedso as to obtain the best possible linear regression. To give an example,in the numerical range 16000-16200 there are 21 prime numbers, andsubstitutions of the compact set 13 leading to one of these primesnumbered 36, so that the number of successes S was 36/21 which isabout 1.714, from which ln S was deduced, about 0.539. Notice that in thatexample the number of successes exceeds that of primes, due to the fact


Table 1In this table the mean number of successes per prime number in

the various numerical ranges investigated is shown as a function

of the compact set qs. The statistics (except for the range K ,

see below) were made in numerical ranges of about 130 prime

numbers, starting at X0 as indicated. The statistics are satisfactory

only for the lesser compact sets, where the number of successes is

relatively large. K is the proportion of successes in the confidence

range for prime qs+1 . . . (qs+1)2 − 1, where s is the rank of the last

prime in the compact set qs. One has for the range of K : 11 :

13 . . . 168; 13 : 17 . . . 288; 17 : 19 . . . 360; 23 : 29 . . . 840;

29 : 31 . . . 960; 31 : 37 . . . 1368; 37 : 41 . . . 1680

Value of X0 11 13 17 19 23 29 31 37K 5.176 4.036 2.662 1.407 0.679 0.296 0.168 0.104

1× 104 1.148 2.224 1.892 1.306 0.681 0.258 0.110 0.0465× 104 0.492 1.110 1.300 1.156 0.790 0.250 0.133 0.043

2.5× 105 0.183 0.474 0.832 0.828 0.706 0.314 0.110 0.0431.25× 106 0.0676 0.122 0.398 0.620 0.465 0.240 0.107 0.0206.25× 106 0.0213 0.088 0.169 0.252 0.252 0.157 0.142 0.057

3.125× 107 0.0083 0.0279 0.052 0.106 0.197 0.125 0.118 0.0751.5625× 108 0.0023 0.0102 0.027 0.055 0.098 0.051 0.069 0.046

that a same prime may be obtained more than once by the substitutionsin the compact set 13. This does not mean, however, that all theprimes in the interval were attained. As X is increased, the phenomenontends to disappear, as multiple successes for a given prime become rareoccurrences.

Figure 1 shows the plot of ln S versus [ln ln X]5 , the best linearregression for the compact set 11.

Figure 2 shows the plot of ln S versus [ln ln X]7.5 , the best linearregression for the compact set 13.

From figures 1 and 2 the following approximate (pseudo statistical)relationships follow, for the rate of success S as a function of X :

S11 ∼ [ln X]−u u ∼ [ln ln X]4 (14)

S13 ∼ [ln X]−v v ∼ [ln ln X]6.5 (15)

Notice that the power of ln ln X in the figures and in eqs. (14) and(15) is not the same. One of course should not expect that the value of S


Figure 1ln S versus [ln ln X]5 for the compact set 11. S is the number of

“hits” or successes per prime and X is the mean of the numerical

interval investigated starting at X0 , X = X0 + ∆X/2. R is the

regression coefficient and SD is the standard deviation

will be exactly given for every value of X by either of the above laws,once the proportionality constants are specified. Deviations from theactual value for a given prime (more generally, integer) may be important.Since the numerical range investigated was limited to 1.5 × 107 for thecompact set 11 and 2 × 107 for the compact set 13, one cannot besure, especially regarding the exponents u and v, that the laws displayedin eqs. (14) and (15) will hold for larger primes (the contrary is more likely).For the purposes of the present paper, however, admitting the universalityof the above laws can by no means invalidate the arguments to follow.

We now introduce the probability that an arbitrary prime P lying inthe range X0 , X0 + ∆X can be represented by the compact set qs. Letthis probability be denoted by W(P, s) < 1. This probability is distinctfrom the rate of successes S displayed in Table 1 and Figure 1-2, becauseof multiple representation of small integers, prime or not, by the lesser


Figure 2ln S versus [ln ln X]5 for the compact set 13. S, X , R and SD

are defined as in Figure 1

compact sets. When however X is increased, the two tend to coincide,as multiple representations become scarce. Now, the probability that theprime P will not be represented by any of the relevant sets, will be:

Ω19(P) =t=s−1

∏t=s0

[1−W19(P, t)] , (16)

where S0 refers to the rank of the BCS of prime P and s − 1 to the rankof the prime preceding P. (The reason of the subscript 19 in Ω(P) andW(P) will become clear in the next paragraph). All elements W19(P, t) ofthe above product are less than one but finite, so that Ω19(p) is also finite.Therefore, there is a finite probability that an arbitrary, not too small, primenumber, will not be attained by any of the substitutions of the relevantcompact sets.

A major objection which can be made to the above argument is thatthe data in Table 1, which may be confused with the probabilities W forX sufficiently large, have been obtained using a computer which retainedinteger numbers having at most 19 digits. Why “hits” will not occur for


A’s in the relationship C = A − B (eq. (5)) having, say, 122 digits, or13201 digits, and so on to infinity? Thus, one may argue that in eq. (16)the possibility that the W(P, s) tend to the value one is not excluded, andtherefore, the product in eq. (16) might in fact be zero.

Again, while not constituting a rigorous mathematical proof, com-puter numerical evidence strongly suggests that this cannot be so. First,considering only positive values for Ck , it is quite clear that in eq. (5)the number of digits in Ak is at least equal to the number of digits inCk . In Table 2 are numbered, for the substitutions in the compact set13, the one thousand least Ck ’s, primes or not, beginning with the Ck

corresponding to the first prime after 103 , 104 and up to 107 . Since ourcomputer could deal with figures having at most 19 digits δ , the computerobtained value for the probability corresponds to

W19(P, s) =19

∑δ=δ0

W19(P, s, δ) . (17)

Table 2Number of digits in A for a given number of digits in C in the

relationship C = A − B for the compact set 13. The data

were compiled from the one thousand least Ck ’s, prime or not,

beginning with the first prime after 10z , z going from three to

seven

Nb of digits in C 4 5 6 7 8Nb of digits in A 103 104 105 106 107

4 803 — — — —5 96 823 — — —6 78 142 919 — —7 18 27 67 937 —8 5 7 8 51 9609 0 1 5 10 3710 0 0 1 2 311 0 0 0 0 0

We are interested in W∞(P, s), the value of the probability when norestriction is imposed to the number of digits δ . Now the data in Table 2allow one to write: W(P, s, δ + 1) = W(P, s, δ)/k and generally

W(P, s, δ + n) = W(P, s, δ)/kn , k > 1 (18)


with the assumption that k is a constant through all the δ range. Eventhough this assumption is only approximate, and may even be wrong asthe number of digits δ is increased, this will not affect the conclusion tofollow, provided that k > 1, whatever the value of δ .

It is seen from Table 2 that the number of cases where A has as manydigits as C is overwhelmingly predominant, and the higher the power of10, the more so. Now by dividing the data in Table 2 by 1000, one obtainsroughly probabilities, so that, with the assumption of a constant k,

W∞(P, s) =∞∑

δ=δ0

W(P, s, δ) = W(P, s, δ0)/dk/(k− 1)e . (19)

Inspection now of Table 1 shows that as soon as P is large enough,W19(P, s) is quite small, W19(P, s) ¿ 1, so that whatever k > 1 in eq. (19),as evidensed in Table 2, one should still have W∞(P, s) ¿ 1, even thoughW∞(P, s) ≥ W19(P, s). Consequently, Ω∞(P) obtained by replacing ineq. (16) W19(P, s) by W∞(P, s) will remain finite as was W19(P, s), beingthe product of finite quantities 1 − W∞(P, s). The conclusion follows,in the form presently of a conjecture: All integer numbers, prime or not,can be divided into two classes: those which can be represented by one or moresubstitutions in compact sets of primes qs, and those which cannot.

This conjecture is based on the euristic argument developed aboveon the number of digits in the ordered substitutions of the compact set13. It is therefore implicitly assumed that the number of digits in Awill follow the same general pattern for other compact sets, especiallywhen qs > 13. This is likely but has not been tested. Also, the generalargument preceding the enunciation of the conjecture lacks mathematicalrigour. Therefore, it is stressed that the conjecture stated above has to beproved or disproved according to the result of a rigorous mathematicaltreatment.

Conclusion

This article, together with the two following appendices, shouldprimarily be viewed as an investigation on some properties of integernumbers. Pending a rigorous mathematical proof to prove or possiblydisprove it, the conjecture is made, based on appropriate numericalevidence, that all integer numbers, prime or not, can be classified into twoclasses: those which can be represented as a sum or difference of products


of primes forming a compact set, and those which cannot. Focusingnow on prime numbers, it is clear that no one will ever imagine thatthe procedure here exposed may actually serve to enlarge the frontierof presently known prime numbers. This would imply working withnumbers in compact sets of the order of the primorial function qs# =2× 3× 5× . . .× qs , where qs is the largest presently known prime forminga compact set with its lesser also known primes. Even the most powerfulcalculators of the future, will presumably be unable to deal with such largenumbers! This however does not dispel the theoretical possibility of sodoing, even though, on practical grounds, the procedure is unattainableand therefore of theoretical only interest.

Today there are relatively efficient methods to test the primality ofan odd integer. This however is quite distinct from ensuring that an oddinteger P, obtained through some substitution in a compact set qs, is,as indicated in the text, either a prime, or a product of at most n primes,depending on the numerical values of P and of qs , the largest prime in thecompact set qs.

Appendix A. Representations of unity through compact sets of primes

We first give below the representations of unity by compact sets ofprimes, as computer found, up to B = 1010 :

1 = 2− 1

1 = 3− 2

1 = 4− 3 = 22 − 3

1 = 6− 5 = 2× 3− 5

1 = 9− 8 = 32 − 23

1 = 10− 9 = 2× 5− 32

1 = 15− 14 = 3× 5− 2× 7

1 = 16− 15 = 24 − 3× 5

1 = 21− 20 = 3× 7− 22 × 5

1 = 25− 24 = 52 − 23 × 3

1 = 36− 35 = 22 × 32 − 5× 7

1 = 81− 80 = 34 − 24 × 5

1 = 126− 125 = 2× 32 × 7− 53


1 = 225− 224 = 32 × 52 − 25 × 7

1 = 385− 384 = 5× 7× 11− 27 × 3

1 = 441− 440 = 32 × 72 − 23 × 5× 11

1 = 540− 539 = 22 × 33 × 5− 72 × 11

1 = 715− 714 = 5× 11× 13− 2× 3× 7× 17

1 = 1716− 1715 = 22 × 3× 11× 13− 5× 73

1 = 2080− 2079 = 25 × 5× 13− 33 × 7× 11

1 = 2401− 2400 = 74 − 25 × 3× 52

1 = 3025− 3024 = 52 × 112 − 24 × 33 × 7

1 = 4375− 4374 = 54 × 7− 2× 37

1 = 9801− 9800 = 34 × 112 − 23 × 52 × 72

1 = 12376− 12375 = 23 × 7× 13× 17− 32 × 53 × 11

1 = 123201− 123200 = 36 × 132 − 26 × 52 × 7× 11

1 = 194481− 194480 = 34 × 74 − 24 × 5× 11× 13× 17

1 = 633556− 633555 = 22 × 7× 113 × 17− 33 × 5× 13× 192

Notice that no other representation of unity was found beyond 1 =633556 − 633555 up to B = 1010 . This suggests the possibility that thenumber of representations of unity through compact sets is finite. If thissuggestion holds (only a theoretical proof could of course firmly establishthe conjecture), then this would have the following consequences:

• First, the number of UCS’s for any integer, prime or compound,will equal the number of UCS’s of unity. This is the immediateconsequence of the fact that in order to obtain an UCS of any integerN , one must multiply both members of a given proper representationof unity by N .

• A second consequence would be the following theorem: If B = Xis the largest representation of unity through a compact set, 1 =(X + 1)− X then no two successive integers beyond B = X can forma compact set of primes

• A third and far more important consequence would by as follows:Though unity has no BCS or OCS’s and is represented throughcompact sets only by UCS’s, the finite number of possibilities to rep-resent unity through compact sets might suggest that the same may


happen for all other primes. If this were so, this would practicallyexclude a positive answer to the question asked in paragraphs 1.6and 2.3 above. That is, there exists a class of primes which cannot bedecomposed in a sum A + B or a difference A− B forming a compactset of primes.

This is in accord with the conclusion given previously. It is howeveronce more stressed that only theoretical demonstrations can definitelyresolve the question one way or the other, suggestions originating incomputer calculations having only putative validity, until a rigorousmathematical proof can be given.

Appendix B. Descriptive presentation of the compact set 5To further illustrate what has already being shortly indicated in the

main text (see 1.5), let us here expose the results obtained by the detailedcalculation of all substitutions (partitions) of the compact set 5 =1, 2, 3, 5 up to the fourth power of its elements. These calculations canbe easily performed using only a small hand calculator.

First, all primes in the range 7 to 48 = 72 − 1 are obtained at leastonce by some combination of 5. 47 = 2 × 52 − 3 was the only primein the range to have being obtained only once; all other primes havebeing obtained at least twice, while 11 and 13 have being spotted fourtimes through various combinations of 5. Of course, this does not meanthat the same will occur for larger compact sets. Indeed, for higher sets,only part of the primes in the corresponding range of confidence will beobtained. Thus, computer calculations have shown that for 37, up to thetenth power for the primes in this compact set, only 17 primes among the251 lying in the range of confidence 38 . . . 1680 are obtained, a proportionof only 6.77%, which is bound to further decrease as qs is increased.

Then, the substitutions lying in the range 49 . . . 342 = 73 − 1, whichmay be either primes P or the product of two primes P1P2 have beenexamined. Among these substitutions, we exclude from the statistics thosewhich are necessarily compound numbers because of algebra, like

119 = 24 × 32 − 52 = a2 − b2 − (a− b)(a + b) = 7× 17,

or else

91 = 23 × 33 − 53 = a3 − b3 = (a− b)[(a− b)2 + 3ab] = 7× 13,


with a = 6 and b = 5. The remaining substitutions where equivocationwas allowed numbered 45.

From these 30 were primes and 15 the product of two primes. Thus,the probability that a substitution leads to a prime is 30/45, which isabout 67%, and consequently the probability of obtaining a product P1P2

is about 33%. Notice that if an integer is chosen at random in the range49 . . . 342, since 53 is the number of primes in that range, the probabilitythat it be a prime is 53/(342− 48), which is about 18%. Notice also, thatonly 27 out of the 53 primes in the range were visited, because primes71, 89, and 191 were visited twice. Thus, the proportion of visited primesis 27/53, that is about 51%. In the same manner, the total number ofproducts P1P2 in the same range is 26. (Products containing either 2, 3,or 5 are naturally excluded from that count, since these cannot be reachedby any of the substitutions of the compact set 5, in accord with thefundamental property of the main text.) Thus, at most 15 out of the 26permitted products P1P2 could have been reached. Actually, these areonly 11, because compound numbers 77 = 7 × 11, 91 = 7 × 13, wereobtained twice, while compound number 119 = 7× 17 was obtained threetimes. Thus, only 11 out of 26 permitted products (which is about 42%)are obtained from the substitutions of the compact set 5 in the rangeindicated.

One should however keep in mind that the count was limited to thefourth power of the primes included in 5; it is clear that considerationof higher powers will modify the statistics here given (see Table 3 below).

Next we come to the third “layer”, 343 . . . 2400 (since 74 − 1 = 2400),where the substitutions may be either primes P, or the product of at mostthree primes P1P2P3 . Actually it was found that among the 43 substitutionsfalling in that range, only 34 × 52 − 2 = 2023 = 7× 172 , was the productof three primes. Of the remaining 42, 28 were primes and 14 the productof two primes.

In the fourth “layer”, 2401 . . . 16806 (16806 = 75 − 1), whereproducts of up to four primes are permitted, 33 substitutions were found,of which 16 were primes and 17 compound numbers. Two, 23 × 54 − 32 =4991 = 7 × 23 × 31 and 24 × 34 × 5 − 1 = 6479 = 11 × 19 × 31, weretriple products, all the other being binary product of primes, so that noproduct of four primes was found.

Finally, in the fifth “layer”, 16807 . . . 117648, 15 substitutions were


found, most of them not being primes.

Table 3Proportion of primes in the range (qs+1)2 . . . (qs+1)3 − 1, for

several values of the compact set qs. In this range, C = A − B

is either prime or the product of two primes. In column four,

the proportion of primes attained with respect to the total in the

numerical range indicated is also shown

Compact set Numerical range Prime C’s % Primes attained %3 5 . . . 124 64.7 52.45 49 . . . 242 74.2 94.77 121 . . . 1330 68.2 97.311 169 . . . 2196 67.5 97.913 289 . . . 4912 66.3 98.317 361 . . . 6858 68.0 91.719 529 . . . 12166 67.3 78.423 841 . . . 24388 66.7 77.1

Now computer calculations up to the twentieth power of the ele-ments in the compact set have shown that the proportion of primes withinthe range (qs+1)2 . . . (qs+1)3 − 1, where Ck may be either a prime or aproduct of two primes, is fairly constant and of the order of 2/3. Table3 gives more detailed results up to the compact set 23.

Notice that for compact set 5 the proportion of primes in the range(qs+1)2 . . . (qs+1)3 − 1 is not exactly that given previously. This shouldbe ascribed to the fact that previously the powers in the elements of thecompact set 5 were limited to the fourth, instead of the twentieth inTable 3. To summarize, primes should be looked for not only within theconfidence interval qs < Ck < (qs+1)2 , but also, with a good probabilityof success, in the range (qs+1)2 < Ck < (qs+1)3 . Higher compact setshave not been investigated because of the computational time involved,but the conjecture is made that the proportion of primes in the range(qs+1)2 . . . (qs+1)3 − 1 will remain fairly constant when qs is increased.Table 3 also shows that the percentage of primes attained goes througha maximum for the compact set 13, but then diminishes and is boundto continue so with increasing qs , as this is also evidenced from the resultsin Table 1.


References

[1] K. Ireland and M. Rosen, A Classical Introduction to Modern NumberTheory, 2nd edition, Springer-Verlag, Berlin — New York, 1990.

[2] J. Itard, Les Nombres Premiers, coll. “Que sais-je?”, Paris, 1969.[3] S. S. Miller and R. Takloo-Bighash, An invitation to Modern Number

Theory, Princeton university Press 2006.[4] P. Ribenboim, The New Book of Prime Number Records, 3rd edition,

Springer-Verlag, New York — Berlin, 1996.[5] H. E. Rose, A Course in Number Theory, 2nd edition, Clarendon Press,

Oxford, 1994.[6] P. Samuel, Theorie algebrique des nombres, 2nd edition, Hermann,

Paris, 1971 (last printing 1997).[7] M. R. Schroeder, Number Theory in Science and Communication,

Springer-Verlag, Berlin — New York, 1990.[8] Ref. [5], p. 7.[9] Ref. [2], p. 27.

[10] Ref. [5], p. 4.

Received December, 2006

a search of primes from lesser primes

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