Certain forms of even numbers and their properties.
Introduction
Some problems of combinatorics have been found in number theory. The number theory has been occupying a field of conjectural problems. Among these conjectural problems (statements), the conjectural problems of prime number has been playing an important role. The scientists in general and number theorist in particular have been thinking since the establishment of the conjectures to get the formal proof of them. Some sophisticated conjectural problems, quite a few of them are twin prime conjecture, mersenne prime conjecture and Goldbach conjecture. The twin prime conjecture states that if p is prime, then there exist infinitely prime pair (p, p+2). Similarly, the mersenne prime conjecture states that if p is prime, then [2.sup.p]1 is also a prime. The Goldbach conjecture [1742] is that every even number larger than 2 can be expressed as a sum of two primes. Besides, the existence of prime between [n.sup.2] and [(n+1).sup.2] is also a conjectural statements. Again it is found that there are many primes of the form 4n+3, 6n+5 and 8n+5 for n [greater than or equal to] 1. Number theorist have been searching various properties of primes which can be actually supposed as a never ending theory. One of the very important non settled problems is to find out a general formula or functional relationship in terms of n where n [greater than or equal to] 1. It has been found that [n.sup.2]n +41 is prime for 1 [less than or equal to] n [less than or equal to] 40 [5]. But the prime numbers obtained from this formula are not consecutive. Another formula [n.sup.2]79n +1601 has also been forwarded for 1 [less than or equal to] n [less than or equal to] 79, from which only 79 primes have been calculated [1]. Recently, in 2000, some new direction of proof of Goldbach conjecture has been forwarded by Kalita [2]. The final proof of Gold bach conjecture for all even numbers greater than 26 has been forwarded by Kalita [3] in 2006. He considered two sets of even numbers {4n+10/ n [greater than or equal to] 4} and {4n+12/ [greater than or equal to] 4}, as their union gives all even numbers [greater than or equal to] 26 and forwarded the proof of Goldbach conjecture in easy and lucid way. He has forwarded the proof of Goldbach conjecture in opposite direction. That is if sum of two primes can be expressed as an even number n [greater than or equal to] 2, then it gives the proof of Goldbach conjecture. It has been found [14] that the presumption is true for all even numbers between 2 and [10.sup.18]. Further, a theorem regarding the existence of consecutive primes [greater than or equal to] 13 has also been developed with some experimental results. Besides, two new conjectures regarding the existence and nonexistence of perfect numbers had been proposed by Kalita. He proposed that all perfect numbers lie in the set {4n+12/ for some particulars values of n [greater than or equal to] 4}, but no perfect numbers exist in the set {4n+10/n [greater than or equal to] 4}. Interestingly, the two sets {4n+10/n [greater than or equal to] 4} and {4n+12/n [greater than or equal to] 4} have many properties. Some of them already have been discussed by kalita [3]. Again it has been found that the pentagonal numbers n(3n1)/2 for n [greater than or equal to] 1 have some structures like a graph but it does not mean a graph as it does not satisfy the basic property of graph theory, that is, sum of the degrees of a graph is equal to the twice of edges. That is if we suppose the pentagonal number 5, then definitely it has a structure but when it is considered as a graph then the structure gives the graphical partition of 10. Some special properties of pentagonal numbers have been found. It will be discussed later that each pair of even pentagonal numbers, one will lie in the {4n+10/n[greater than or equal to] 4} and other will lie in {4n+12/ n[greater than or equal to] 4}. Some relations of partition and graphical partition have been discussed in [11] and Rousseau and Ali [12] showed it as Limit n [right arrow] [??] g(n)/ p(n) [less than or equal to]. 25. Another important relation of graphical basis partition h(n) of n, basis partition b(n) of n, graphical partition g(n) of n and partition p(n) of n had been found as Limit n [right arrow] [??] (n)/ b(n) = Limit n [right arrow] [??] g(n)/ p(n). In this paper, we have been forwarding some theoretical aspects of the even numbers of the sets {4n+10/n[greater than or equal to] 4} and {4n+12/ n[greater than or equal to] 4}. Some properties of pentagonal numbers and the relationship of partition functions of numbers have also been discussed. Further, some properties and a few open questions regarding the some kind of even numbers have been proposed. Before going to discuss the main topics we remind some definitions related to even numbers [3]. 2. Definitions:[3] 2.1. Good Numbers: The numbers, which are obtained from the set {4n+10/n[greater than or equal to] 4} but not multiple of 3,5,7 and 11 are called Good Numbers. The set of Good Numbers is denoted by GNS. 2.2. EN1 Numbers: The Good Numbers of the form 2[p.sup.i] for i[greater than or equal to] 2 for the primes [greater than or equal to] 13 are called the exceptional class one numbers and their set is denoted by EN1. 2.3. EN2 Numbers: The Good Numbers of the form 2[P.sub.1][P.sub.2][P.sub.3].... where [P.sub.1] [not equal to], [P.sub.2] [not equal to] [P.sub.3] [not equal to],.... are called exceptional class two numbers and their set is denoted by EN2. 2.4. VEN Numbers: The Good Numbers of the form 1334, 13334, 133334, 1333334,.... are called Very exceptional Numbers and their set is denoted by VEN. 2.5. VVEN Numbers: The Good Numbers of the forms 1222, 122222, 1222222, are called Very Very exceptional Numbers and their set is denoted by VVEN. Remark 1: Some remarks of VVEN numbers have been stated in [3] regarding the form of 1222,122222, 1222222 ... that is, the present of the digits 2 should not be considered for the above numbers when we define for 3n+1 n[greater than or equal to] 1 where n is the numbers of 2. The following theorems are very important of some even numbers. Theorem 1: The even numbers which are expressed in the forms (i) [i.sup.3] +2i for i[greater than or equal to] 2j for j[greater than or equal to] 2, (ii) i4 4i2 for i[greater than or equal to] 2j for j[greater than or equal to] 2, are always lie in the set {4n+12/n[greater than or equal to] 4}. Proof: We first prove the statement (i) of the theorem and the statement (ii) will follow the proof of statement (i). According to the definition of the two sets {4n+10/n[greater than or equal to] 4} and {4n+12/n[greater than or equal to] 4} it has been found that their union gives even numbers [greater than or equal to] 26. We now show that the even number [i.sup.3]+2i for i[greater than or equal to] 2j, j[greater than or equal to] 2 will not lie in the set {4n+10/n[greater than or equal to] 4}. If possible, without loss of generality, we suppose that the number [i.sup.3]+2i for i[greater than or equal to] 2j, j[greater than or equal to] 2 belongs to the set {4n+10/n[greater than or equal to] 4}. Then we can immediately suppose that this number is a Good Numbers or any even numbers of the set. Suppose this number is a Good number. Then for j[greater than or equal to] 2 the number 8[j.sup.3]+4j is not multiple of 3,5,7 and 11 (by definition). But, we see that the number 8[j.sup.3]+ 4j for j [greater than or equal to] 2 is divisible by 3. Hence the number 8[j.sup.3]+4j for j [greater than or equal to] 2 is not a Good Number. Similarly we can show that the number [i.sup.3] +2i for i[greater than or equal to] 2j, j [greater than or equal to] 2 is not EN1, EN2, VEN AND VVEN. Finally we observe, if [i.sup.3]+2i for i[greater than or equal to] 2j, j [greater than or equal to] 2 belong to the set {4n+10/n[greater than or equal to] 4}  {GNS U EN1 U EN1 U VEN U VVEN}, then for any value of n[greater than or equal to] 4 and j [greater than or equal to] 2 the value of the expression 4n+10 and 8[j.sup.3]+4j must be equal. But, we observe that, when j=2 then 8[j.sup.3]+4j=72, but there does not have the value 72 for 4n+10 for n [greater than or equal to] 4. Hence we have the number 8[j.sup.3]+4j for j [greater than or equal to] 2 does not lie in the set {4n+10/n [greater than or equal to] 4}. Therefore our supposition is wrong. Hence 8[j.sup.3]+4j for j [greater than or equal to] 2 that is i3 +2i for i[greater than or equal to] 2j for j[greater than or equal to] 2 lie in the set {4n+12/ n[greater than or equal to] 4] and which completes the proof. Remark 2: It has been proposed [3] that all perfect numbers lie in the set {4n+12/ for some particular values of n [greater than or equal to] 4}. But, here, the even numbers of the theorem [1] above does not give any perfect numbers. Hence we can leave these numbers for testing the existence perfect numbers as discussed in [3]. Remarks 3: The author has considered the statement of the theorem [1], when he studied the problems related to some kinds of numbers [problems no 1.34 and 1.35 of [13]]. The statement of the theorem gives some different ideas of certain numbers. Because he found that the numbers obtained from the theorem do not lie in the set {4n+10/ n [greater than or equal to] 4}. Theorem 2: No number of the set {4n+10/n[greater than or equal to] 4} can be expressed as an even number as stated in theorem no.1. Proof: We can prove it by contradiction process as discussed in the proof of the theorem.1. Theorem 3:The equation 4n+8=2p is not solvable for the prime p[greater than or equal to] 13 for any values of n[greater than or equal to] 4. Proof: We prove it by contradiction process. We suppose that the equation 4n+8=2p is solvable for n [greater than or equal to] 4 with simultaneous changes of the consecutive primes [greater than or equal to] 13. Hence we can write the equation 4n+8=2p as 4n  2p=  8 which is nothing but a diophantine equation. There fore this equation has integer solution as g.c.d (4,2) divides 8. Hence the set of solution of all integers pair (n,p) are found to be n= 1  2t, t=....2,1,0,1,2,3,..... p= 6  4t, t=....2,1,0,1,2,3,.... where the pair (1,6) is initial solution. It is observed that for any values of t=....2,1, 0, 1, 2,.... we have some values of n and accordingly we have some values of p. But when put the value of n for some values of t we have equal value of left hand and right hand side of the equation but p will not be prime. (That is when we put say n = 5 (t=  2), then p equals 14 which is not prime. Hence our supposition that the equation is solvable for n [greater than or equal to] 4 for simultaneous changes of prime [greater than or equal to] 13 is wrong. This completes the proof. Theorem 4: The equation 4n+ 10=2p is solvable for the prime values [greater than or equal to] 13 and for some values of n[greater than or equal to] 4. Proof: We can prove it as discussed in theorem 3. Theorem 5: In every pair of even pentagonal numbers (pen1,pe[n.sup.2]), one lies in the set {4n+10/n[greater than or equal to] 4} and the other lies in the set {4n+12/n[greater than or equal to] 4} Proof: We know that the general form of pentagonal numbers is n(3n1)/2 for n [greater than or equal to] 1. It has also been found that for all values n[greater than or equal to] 1, the pentagonal numbers are nothing but a pair of odd and even numbers. [ for example, the pairs (1,5), (12, 22), (35, 51) etc are found for n=1 to 6 ]. Hence the pair of even numbers are automatically must lie either in {4n+10/n[greater than or equal to] 4} or {4n+12/n[greater than or equal to] 4}, since the union of these two sets gives the even numbers [greater than or equal to] 26, hence we forward the proof of the theorem for even numbers [greater than or equal to] 26. If we suppose that any one of the even pentagonal numbers of pen1 and pe[n.sup.2] lie in both the sets {4n+10/n[greater than or equal to] 4} and {4n+12/n[greater than or equal to] 4}, then there must have some values of n[greater than or equal to] 4 which give these two even numbers. But this will not possible for these two even numbers. For example when we put n=7 and 8 we have two even pentagonal numbers and they are 70 and 92 which are obtained from n(3n1)/2 for n [greater than or equal to] 1. But 70 and 92 are obtained for n=15 in {4n+10/n[greater than or equal to] 4} and n=20 in {4n+12/n [greater than or equal to] 4}. Similar cases will lead that both the even pentagonal numbers will not lie in the both the sets. Theorem 6: The difference between two even pentagonal numbers of each pair is 12n2 for n [greater than or equal to] 1. Proof: We prove the theorem by induction method. The theorem is true for n=1. That is, the difference between the first pair of even pentagonal numbers shown above is 10 (2212=10). Let us suppose that the theorem is true for n=k. Hence the difference between two pentagonal numbers is 12k2. We now show that it is true for n=k+1. When we put n=k+1, then the difference is 12(k+1)2=12k+10. But our theorem states that the theorem is true for n [greater than or equal to] 1, which implies that the theorem is rue for n=k+1 [greater than or equal to] 1, implies k [greater than or equal to] 0. This completes the proof. Theorem 7: The difference between the odd pentagonal numbers of each pair is 12n8 for n [greater than or equal to] 1. Proof: Proof follows the proof of the theorem 6. Theorem 8: The sum of two pentagonal numbers (even & odd) is not a pentagonal numbers but sum of even pentagonal numbers of consecutive pair lies in the set {4n+10/ n [greater than or equal to] 4}, but does not lie in the set {4n+12/ n [greater than or equal to] 4}. Proof: We know that the sum of two even numbers is again an even number and the set {4n+10/ n [greater than or equal to] 4} is also a set of even numbers and hence it can be proved immediately. Remark3: But the sum of two odd pentagonal numbers interestingly lie in the set {4n+10/n[greater than or equal to] 4} The following three results are proposed which seem to be very important with partition of numbers, graphical partitions and perfect numbers related to the sets {4n+10/ n[greater than or equal to] 4} and {4n+12/ n [greater than or equal to] 4}. Proposition 1: Limit n [right arrow] [??]g(4n+10)/ g (4n+14) = limit n [right arrow] [??] p(4n+10)/ p(4n+14)=1 Proposition 2]: Limit n [right arrow] [??] g(4n+12)/ g(4n+16) = limit n [right arrow] [??] p(4n+12)/ p(4n+16)=1 Proof: Various types of even numbers are obtained from the sets {4n+10/n[greater than or equal to] 4 }, {4n +14/ n[greater than or equal to] 4}, {4n+12/ n [greater than or equal to] 4} and { 4n+16/ n [greater than or equal to] 4} and we can test the propositions for them which will show that the propositions are true. The value of graphical partition and partition of the number 94 are shown below as g(94) / g(98)= .618037285 p(94) / p (98)=.616983156, which is tending to 1. For other numbers also we can find the limit of partition and graphical partition which will automatically show the limiting values of propositions 1 & 2 as 1 when n lends to [??]. Proposition 3: Some pentagonal numbers are perfect numbers. Proof: Since some pentagonal numbers lies in the set { 4n+12/n [greater than or equal to] 4} and it has been already shown in [3] that all perfect numbers lie in the set {4n+12/[greater than or equal to] 4 }. Hence the pentagonal numbers are perfect numbers. Proposition 4: The numbers 202,20002,2000002, 200000002,.... lie in the set {4n+10/ for some particular values of n [greater than or equal to] 4} and they are not multiple of 3, 5, 7 and 11 and hence 101, 10001, 1000001 100000001,.... are primes. Proposition 5: The numbers 262, 26662, 2666662, 266666662,....lie in the set {4n+10/ for some particular values of n [greater than or equal to] 4} and they are not multiple of 3, 5, 7, and 11 and hence 131, 13331, 1333331, 133333331,....are primes. Proposition 6: The numbers 422, 4222,42222, 422222,....lie in the set {4n+10/ for some particulars values of n =4} and they are not multiple of 3,5,7 and 11 and hence 211, 2111, 21111, 211111,....are primes. Some open questions: (1) What is the largest even number of one core digits whose last four digits and first three of the number begin with 1,3,6,9 and 1,9,7. (2) What is the largest even number of one/two/three lakhs digits of 3,4,5,6.. Conclusion: The paper indicates only some properties of the sets {4n+10/n [greater than or equal to] 4} and {4n+12/n[greater than or equal to] 4} which are all even numbers. There are so many open problems on these two sets of even numbers which will focus in near future some unsolved of prime numbers. References [1] G.H. Hardy and E.M Wright. "An introduction to the theory of Numbers" Oxford U.Press, 1981. [2] Kalita, B. "Some conjectural problems and partial solution of it" Proc. ICRAMS, 2000, I.I.T, Kharagpur, dec 2022. [3] Kalita, B, "Proof of Gold bach conjecture and related conjecture of primes" Bull. Pure & Applied Sciences, Vol25(E)No2, pp451459, 2006. [4] Courant, R and Robbins, H. "What is mathematics" Oxford Uni. Press, 1996. [5] Loweke, G.P, "The lore of prime numbers" Vontage press, N.Y. 1982. [6] Pomerance, C. " The search for prime numbers" Scientific American.249, 1982. [7] Andrews. G.E. "A combinational proof of a partition function limit" Amer. Math. Monthly. 78, 1971, 276278. [8] Erdos, P, Montgomery, H.L " Sums of numbers with many divisors" J.Number theory, Vol. 75 (1), 1999. 16. [9] Yokota, H. "The largest integer expressible as a sum of reciprocal of integers" J. N. Theory. Vol. 76(2), 1999. [10] Jenkner. W. "On divisors whose sum is a square" Acta, Arithmetica, Vol 90(2), 1999, pp113120. [11] Jennifer, M. et al " Graphical basis partitions" Graphs and Combinatorics, 1998,14 pp241261. [12] Rousseau & Ali " On a conjecture concerning graphical partitions, congruent number 104,1994, Juornal of com. Theory Series, 1994, pp 150160. [13] C. L. Lie, Elements of District Mathematics, Mc Graw Hills, 1985. [14] Alex van d et al. "Goldbach suspect : a step further" Source : Kennislink, language: Dutch, 27/4/2007. Bichitra Kalita Department of M.C.A (Computer Application), Assam Engineering College, Guwahati781013, Jalukbari Assam. India, Email: bichitra1_kalita@rediff.com 

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