[PDF] Top 20 Solutions of the Diophantine Equation $2^x + p^y = z^2$ When $p$ is Prime
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Solutions of the Diophantine Equation $2^x + p^y = z^2$ When $p$ is Prime
... of Diophantine equations is ancient, vast, and no general method exists to decide whether a given Diophantine equation has any solutions, or how many ... See full document
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All the Solutions to an Open Problem of S. Chotchaisthit on the Diophantine Equation 2^x + p^y = z^2 when p are Particular Primes and y = 1
... finding solutions of the title equation where p is prime in general is still an open ...integer solutions of the equation for the particular primes p = 7, 13, 29, 37, ... See full document
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All the Solutions of the Diophantine Equations (p + 1)^x – p^y = z^2 and p^y - (p + 1)^x = z^2 when p is Prime and x + y = 2, 3, 4
... The field of Diophantine equations is ancient, vast and no general method exists to decide whether a given Diophantine equation has any solutions, or how many solutions.. The literatur[r] ... See full document
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All the Solutions of the Diophantine Equations $p^x + p^y = z^2$ and $p^x - p^y = z^2$ when p≥2 is Prime
... numbers is infinite. This proof is considered today as one of the simplest and most elementary proofs, but also as one of the classical most beautiful and elegant proofs ever. The simplicity which characterizes our ... See full document
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On Solutions to the Diophantine Equation $p^x + q^y = z^4$
... 2 2n + q = z 4 (3) implying that q = z 4 - (2 n ) 2 = (z 2 – 2 n )(z 2 + 2 n ...is prime, then z 2 – 2 n = 1 and ... See full document
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On the Diophantine Equation ${(q^2 )^n }^x+p^y= z^2$ where q is any Prime Number and p is an Odd Prime Number
... and p is an odd ...any prime number and p is an odd prime number. Some solutions of these Diophantine equations have been ... See full document
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On Solutions of the Diophantine Equations $p^3 + q^3 = z^2$ and $p^3 - q^3 = z^2$ when p, q are Primes
... p 3 - 2 3 = (p - 2)(p 2 +2p + 4) = z 2 ...If p is prime, then the equation p 3 - 2 3 = z 2 has no ... See full document
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On Solutions to the Diophantine Equation M^x+ (M + 6)^y = z^2 when M = 6N + 5
... A prime gap is the difference between two consecutive primes. Articles as [4, 6] and many others have been written on prime gaps. In 1849, A. de Polignac conjectured that for every positive integer k, ... See full document
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On the Diophantine Equation $p^x + q^y = z^2$
... If p is a Sophie Germain prime, then by definition q = 2p+1 is also ...primes p = 2 and p = 3 are also Sophie Germain ...Germain prime p ≥ 2 satisfies the above ... See full document
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On Solutions of the Diophantine Equation $p^x + q^y = z^2$
... Remark 2.1. It is noted, that clearly Lemma 2.1 can not guarantee that 6N + 5, 6N + 7 are both primes. However, it does guarantee that for each value of R when R = 1, 2, … , the right-hand side of ... See full document
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Solutions of the Diophantine Equation $p^x + (p+6)^y = z^2$ when $p, (p + 6)$ are Primes and $x + y = 2, 3, 4$
... A prime gap is the difference between two consecutive ...on prime gaps, a very minute fraction of which is brought [5, 6] ...primes p such that p + 2k is prime ... See full document
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On the Cubic Diophantine Equation with Four Unknowns $x^2+y^2=z^3-w^3$
... integer solutions to the cubic equation with four unknowns given by x 2 + y 2 = z 3 − w 3 ...As Diophantine equations are rich in variety due to their definition, ... See full document
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On Solutions of the Diophantine Equation $A^2-B^2 = Z^4$ when $A, B, Z$ are Positive Integers
... B, Z. We establish: (i) For all primes A, B, the equation has a unique ...(ii) When B = 4N + 3 (N > 0) is prime, the equation has no ...1 prime, the necessary and sufficient ... See full document
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On the Ternary Quadratic Diophantine Equation $3(X^2+Y^2)-5XY=75 Z^2$
... quadratic equation given by 3 ( X 2 + Y 2 ) − 5 XY = 75 Z 2 is considered and searched for its many different integer ...integer solutions to the above ... See full document
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A Ternary Quadratic Diophantine Equation $x^2+y^2=65z^2$
... integer solutions to the equation given by x 2 + y 2 = 65z 2 ...quadratic Diophantine equations and determine their integer solutions along with suitable ... See full document
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On the Cubic Diophantine Equation with Five Unknowns x3 + y3 + (x+y)(x-y)2=32(z+w)p2
... of Diophantine equations offers a rich variety of fascinating ...cubic equation with five unknowns given by x 3 y 3 ( x y ) ( x y ) 2 32 ( z ... See full document
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On the Non-Homogeneous Quadratic Equation with Five Unknowns $x^2+xy-y^2-(z+w)=10 p^2$
... (i) y ( ) ( ) n − x n (ii) 3 [ p ( ) ( ) n , n − y n ] ...[ z ( ) ( ) α , β + w α , β ] = 5 y ( ) ( ) 4 β 2 − 2 x 10 α 2 ...3. y ( ) ( ) ( ) n ... See full document
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Integral Solutions of Homogeneous Biquardratic Equations with Five Unknowns $2(x^4-y^4)=(z^2-w^2)p^2$
... of Diophantine Equations offer a rich variety of fascinating problems ...biquadratic Diophantine homogeneous and non-homogeneous have aroused the interest of numerous ...integral solutions in its ... See full document
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On The Positive Pell Equation $y^2=90 x^2+31$
... integer solutions for the positive pell equation y 2 = 90 x 2 + 31 ...quadratic Diophantine equations are rich in variety, one may search for the other choices of pell ... See full document
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On the Non-Homogeneous Ternary Quadratic Equation $2(x^2+y^2)-3xy+(x+y)+1=z^2$
... 2 x + y − xy + x + y + = z is studied for its non-zero distinct integer ...integer solutions to the above equation are ...the solutions and special polygonal ... See full document
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