[PDF] Top 20 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
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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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Solutions of the Diophantine Equation $2^x + p^y = z^2$ When $p$ is Prime
... numbers 2 n – 1 being primes dates to antiquity. When n is composite, 2 n – 1 is not a ...that 2 n – 1 is a prime. The first Mersenne Primes [4] are ... 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 Solutions of the Diophantine Equations $p^3 + q^3 = z^2$ and $p^3 - q^3 = z^2$ when p, q are Primes
... p - 2 = 3A 2 and p 2 +2p + 4 = 3B 2 (5) which must exist ...equation p 3 - 2 3 = z 2 has no solutions as ... See full document
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On Solutions to the Diophantine Equation $p^x + q^y = z^4$
... many solutions to equation ...≥ 1, it therefore follows that equation (3) has infinitely many solutions for each and every value n ≥ 1 in which q is composite as ... 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 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 equations, homogeneous and non-homogeneous have aroused the interest of numerous mathematicians since antiquity (Dickson,1952; ... See full document
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Integer Solution of the Homogeneous Bi-Quadratic Diophantine Equation with Five Unknowns $(x-y)(x^3-y^3)=(z^2-w^2)p^2$
... Bi-quadratic Diophantine equations, homogeneous and non -homogeneous, have aroused the interest of numerous mathematicians since ambiguity as can be seen from [1-2] particularly In ... 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 ...] 1 − 3 . In ... See full document
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On Solutions to the Diophantine Equation 3^x + q^y = z^2
... prime, x, y, z are positive integers and x + y = 2, 3, ...4. When q > 3, the cases of infinitely many solutions, of a unique solution ... 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 the Non-Homogeneous Quintic Equation with Five Unknowns $X^4-Y^4=10 p^3 (Z^2-W^2)$
... of Diophantine equations offers a rich variety of fascinating problems ...quintic equations with three unknowns are studied for their integral ...quintic equations with four unknowns for their ... See full document
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On the Non-Homogeneous Ternary Quadratic Equation $2(x^2+y^2)-3xy+(x+y)+1=z^2$
... The Diophantine equation offer an unlimited field for due to their variety [ ] 1 − 3 ...[ 4 − 14 ] for quadratic equations with three ...by 2 ( x 2 + y ... See full document
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Observation on the Cubic Diophantine Equation with Four Unknowns $(x^3+y^3)+(x+y)(x+y+1)=zw^2$
... homogeneous Diophantine cubic equation is an interesting concept as it can be seen from [ 1 , 2 , 3 ] ...[ 4 − 11 ] a few special cases of cubic Diophantine equation with four ... See full document
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All the Solutions of the Diophantine Equation $p^3 + q^2 = z^3$
... the Diophantine equation x n + y n = z n , with integral n > 2, has no solutions in positive integers x, y, ...equation p 3 + q 3 = 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. ... See full document
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On the Ternary Quadratic Diophantine Equation $3(X^2+Y^2)-5XY=75 Z^2$
... follows: 1. X ( ) α , 1 − t 16 , α + 77 ≡ 0 ( mod 11 ) 2. 9 Z ( ) ( ) α , 1 + Y α , 1 − t 38 , α ≡ 0 ( mod 23 ) ...7 Z ( ) ( ) α , 1 + X α , ... See full document
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Integral Solutions of an Infinite Cone α(x^2+y^2 )=(2α 1)xy +(4α 1) z^2
... the solutions in integers of a polynomial equation ( , , … ) = 0, called the Diophantine ...fewer equations than unknown variables and involve integers that work correctly for all ...Quadratic ... See full document
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On the Cubic Diophantine Equation with Four Unknowns $x^2+y^2=z^3-w^3$
... The Diophantine equation offers an unlimited field for research due to their variety ...cubic equations with four unknowns are studied for its non-trivial integral solutions and in [16,17] cubic ... See full document
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On Solutions to the Diophantine Equation M^x+ (M + 6)^y = z^2 when M = 6N + 5
... investigate solutions to the title equation. We establish: (i) For all values M and even values x, y, then the equation has no ...(ii) When M, M + 6 are primes, and x, y ... See full document
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