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On Erdős-Straus Conjecture For IJCSNS International Journal of Computer Science and Network Security, VOL.19 No.3, March 2019 147 On Erdős-Straus Conjecture for Abdul Hameed1, Ali Dino Jumani1, Israr Ahmed1, Inayatullah Soomro1, Abdul Majid2, Ghulam Abbass1, Abdul Naeem kalhoro1. 1Department of Mathematics, Shah Abdul Latif University Khairpur, pakistan 2Department of Biochemistry, Shah Abdul Latif University Khairpur, pakistan Abstract type decomposed as sum of two and three fractions Number theory is very fascinating field of the Mathematics. It has been2 found in Rhind Papyrus scripts. has extensive usage in our daily life. Due to this it has attracted 2+1 research community since the ancient times. Egyptians were very As it is well established that every rational number can much interested in the Number theory. In particular, they made a be expressed as a finite sum of different great contribution towards solving fraction problems. In this 4 unit = + ℕ fractions. For this ( ) study we focus on A conjecture due to Paul Erdos and E.G. 1 1 1 verification and the fraction 4 1 1 1 +1 +1 ∀ ∈ Straus that the Diophantine equation = + + can be written as the sum of two fraction namely as n x y z = + and = + ℕ ( ) ( ) involving Egyptian fractions always can be solved. By using the 2 1 1 4 1 1 for even number and odd numbers of the form = concept of this conjecture we have found that the Erdos-Struass 2−1 2−1 4−1 4−1 ∀ ∈ 3 1 1 1 4 − 1 respectively. conjecture is also valid for = + + . A famous conjecture due to Paul Erdős and E. G. Straus, n x y z known as Erdős-Straus conjecture on Key words: Egyptian fraction states that given a positive integer ≥ 2 Math, Mathematic, Egyptian fractions, Number theory 4 1 1 1 there exists , , such that = + + . n x y z 1. Introduction Here we say that is∈ Erdős ℕ -Straus number and induces a decomposition of the fraction into the sum of three Number theory has remained a very interesting subject because equivalent fractions. Sierpiński and4 Schinzel extended the of its wider applicability in the different fields. This subject is concerned with the study of the numbers. The main applications decomposition of the fraction to a more general are included in the field of computer science, numerical methods, conjecture by replacing 4 by other fixed positive integer information technology, communications, and cryptography. In ≥ 4. particular, Number theory has played a pivotal role in helping to develop algorithms, such as a, RSA a public key cryptography In this note we prove that the fraction can be which considers the fact that there is the difficulty to factor large decomposed into the sum of three Egyptian 3 numbers [1-10]. 3 1 1 1 Diophantine equations are very important topic in the field of fractions, = + + . as well. Number theory. In this topic only integer solutions are allowed. n x y z Diophantine equations are concerned with the Egyptian fraction [4]. The concept of Egyptian fractions has gained attraction of many people going back to the three thousand years and still continues 2. The Proof of the Erdős-Straus Conjecture to attract the interest of mathematicians today. This concept is For basically about the fractions in which numerator is considered as 1 and the denominator appears to be a positive integer [11-16]. Egyptian, in their early time of Second Intermediate Period, In this note we express as a sum of at most three distinct never had a system to write the 3 unit fractions as in the light of Erdos-Straus conjecture. fractions like Instead they write fractions as sum of For all and for some unit fractions like , that is, is to be xyz, ,∈{ 1,2,3,........} 1 11 . written as the sum of and 30 fractions. These fraction has been attracting attention1 since1 the second Intermediate 5 6 Period of Egypt, that is, 1650-1550 BC. Fraction of the Manuscript received March 5, 2019 Manuscript revised March 20, 2019 148 IJCSNS International Journal of Computer Science and Network Security, VOL.19 No.3, March 2019 3111 on the strength of [1] for all n∈∈{ 3,6,9,12...} and for some x,y,z{ 1,2,3,....} Theorem 1. =++, {2, 3, 4, … } nxyz 3111 2n = ++ when x = 2n ∧ y = n ∧ z = n23 nn2n and , , {1, 2, 3, … }. ∀ ∈ 3 Proof. For all {2, 3, 4, … } and for some ∀∈n {6,9,12,15,18,24,...}and for some x, y , z ∈{ 1,2,3,...} ∃ ∈ xyz, ,∈{ 1,2,3........} 31 1 1 4n 4n ∀ =++ when x = n ∧∧ y = z = 3111 nn44nn 26 =++⇒3(xyz = n xy + xz + yz )26 nxyz ∀∈n {10,15,20,25,...}and for some x, y , z ∈{ 1,2,3,...} for n =2 and x =1, y = 4, z = 4 3111 =++ nxyz 3 1 1 1 4n 7n =++ when x = n ∧ y = ∧ z = ∀∈n {3,5,7,11,...} ,for some x,y,z∈{ 1,2,3,...} nn47nn 55 55 3 1 1 1 n++1 nn ( 1) ∀∈n {7,14,21,....}and for some x, y , z ∈{ 1,2,3,...} =++ when x = n ∧ y = ∧ z = nnn +1 nn( +1) 22 2 2 3 1 1 1 4n 28n From (1) it follows that {2, 3, 5, 7…} and for some =++ when x = n, y = , z = m, x, y, z, {1, 2, 3 …} nn4nn 28 77 ∈ 77 (3m−∈ 1) xz n = ∧=∧nm y gcd (n,3m-1) ≥ 1 (2) m(x+ z ) {2}UU{ 3,5,7,11,13,...} { 4,8,12,16,20,...} U{ 6,9,12,15,18,...} U {10,15,20,25,30,...}UU{ 7,14,21,28,35,...} { 3,6,9,12,...} = { 2,3,4,5,6,7,8,...} ∀∈n {4,8,12,16,...} ,for some x,y,z∈{ 1,2,3,...} Obviously{φφ} ≠ . Corollary 1. For all n∈∈{ 2,3,5,7,...}and for some m,x,y,z{ 1,2,3,....} (31m− ) xz n = ∧nm =∧ ygcd( n ,3 m −≥ 1) 1 . mx( + z) 31 1 1 3nn 6 =++ when x = n ∧ y = ∧ z = Which is a required proof of Erdős-Straus Conjecture for nn36nn 44 3 1 1 1 44 = + + . On the strength of (2) =3 and for some x, y, z, {1, n x y z 2, 3 …} ∈ References (31m− ) xz 2xz [1] Culpin D., Griffiths D.,. "Egyptian Fractions." nm= y,11 ∧=∧=∧= m x n = ⇒ Mathematical Gazette (1979): 49-51. mxz()++ xz [2] Elsholtz, C.,. "Sums of k unit fractions." Trans. Amer. Math. 2xz Soc (2001): 353: 3209-3227. (3n== ∧=+ xkz) ⇒ [3] Erdos, P. "On the integer solutions of the equation xz+ 1/x(1)+1/x(2)+...+1/x(n)=a/b." Mat Lapok 1 (1950): 192- 210. 2 [4] Monks M., Velingker A., On the Erd˝os-Straus Conjecture: 2z+( 26 k −) zk −=⇒ 30 Properties of Solutions to its Underlying Diophantine ∆=2 +22 = ⇒ = Equation, Siemens Westinghouse Competition, 2004- 2005. (2kk) 6 10 4 [5] Graham R. L. "Paul Eros and Egyptian Fractions." Janos Therefore Bolyai Society Mathematical Society and Springer-Verlag (2013): 289-309. ( z=∧=+==2 x 4 z 62 nz ∧= 2 n) [6] Jaroma J H. "On Expanding 4/n into Three Egyptian Fractions." Canadian Mathematical Society (2004): 36-37. [7] F. Rogers. “ Fuzzy linear congruence and linear fuzzy integers.” International journal of advanced and applied sicences 3 (6) 2016:10-13 IJCSNS International Journal of Computer Science and Network Security, VOL.19 No.3, March 2019 149 [8] Havare O.H. “ The volken born integral of the p-adic gamma function“ international journal of advanced and applied sciences. 5(2) 2018: 56-59 [9] A.A. Behrozpoor, A.V Kamyad, M. M.Mazarei. “Numerical solution of fuzzy initial value problem (FIVP) using optimization”. international journal of advanced and applied sciences. 3(8) 2016: 36-42. [10] E. Agyuz, M. Acikgoz. “On the study of modified (p,q)- Bernstein polynomials and their applications.” international journal of advanced and applied sciences. 5(1) 2018: 61-65 [11] Ionascu, E. J., A. Wilson, On the Erdos–Straus Conjecture, ˝ Rev. Roumaine Math. Pures Appl., Vol. 56, 2011, No. 1, 21–30. [12] Martin, G.,. "Dense Egyptian fractions." Trans. Amer. Math. Soc. (1999): 351, 3641-3657 . [13] Schinzel, A.,. "On sums of three unit fractions with polynomial denominators. ." Funct. Approx. Comment. Math. (2000): 28:187-194. [14] Sierpiński, W. "Sur les décompositions de nombres rationnels en fractions primaires." Mathesis 65 (1956): 16- 32. [15] Vaughan, R. "On a problem of Erdös, Straus and Schinzel." Mathematika (1970): 17(2), 193-198. doi:10.1112/S0025579300002886. [16] Yamamoto, K. "On Diophantine Equation 4/n=1/x+1/y+1/z." Mem. Fac. Sci. Kyushu U (1965): Ser A. 37-47. .
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