In this paper, we consider the non-linear Diophantine equation π π₯ + (π + 4 π) π¦ = π§ 2 , where π > 3, π + 4 π are primes, x, y and z are nonnegative integers and n is a natural number. It is shown that this non-linear Diophantine equation has no solution.
In this paper, we proved that the Diophantine equation + + = has no solution in non-negative integers x, y, z where p is an odd prime and m, n is a natural number.
This study presents proof and solutions of all non-negative integrals of the Diophantine equation β«β¬ ΰ―« αΊ2β«β¬ ΰ΅ 1α» ΰ―¬ ΰ΅ β«έβ¬ ΰ¬Ά when β«β¬ and β«2β¬ ΰ΅ 1 are primes. This proof uses the prediction of Catalan's conjecture and related theories in proving by separating β«β¬ ΰ΅ 2, β«β¬ ΰ΅ 3 and β«β¬ 3. It shows that if β«β¬ ΰ΅ 2, the Diophantine equation β«β¬ ΰ―« αΊ2β«β¬ ΰ΅ 1α» ΰ―¬ ΰ΅ β«έβ¬ ΰ¬Ά is in the form of 2 ΰ―« 3 ΰ―¬ ΰ΅ β«έβ¬ ΰ¬Ά which has three solutions; αΊ0,1,2α», αΊ3,0,3α» and αΊ4,2,5α» and if β«β¬ ΰ΅ 3 then αΊ1,0,2α» is the solutions of equation 3 ΰ―« 5 ΰ―¬ ΰ΅ β«έβ¬ ΰ¬Ά when considering the value of β«β¬ 3, the Diophantine equation β«β¬ ΰ―« αΊ2β«β¬ ΰ΅ 1α» ΰ―¬ ΰ΅ β«έβ¬ ΰ¬Ά has no solutions.
In this paper, we consider the Diophantine equation (P+ 12) X+ (P+2K)=Z2where p > 3, p are primes and k is natural number, when x, y and z are non-negative integers. It is found that the Diophantine equation has no nonnegative integer solution.
The Diophantine equation has been studied by many researchers in number theory because it helps in solving variety of complicated puzzle problems. From several studies, many interesting proofs have been found. In this paper, the researcher has examined the solutions of Diophantine equation (π΄π β π) π + (π΄π + π) π = π π where π΄π is a Mersenne Prime and p is an odd prime whereas x, y and z are nonnegative integers. It was found that this Diophantine equation has no solution.
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