A conjecture concerning the exponential diophantine equation ax+by=cz
نویسندگان
چکیده
منابع مشابه
On the Exponential Diophantine Equation
Let a, b, c be fixed positive integers satisfying a2 + ab + b2 = c with gcd(a, b) = 1. We show that the Diophantine equation a2x+axby+b2y = cz has only the positive integer solution (x, y, z) = (1, 1, 1) under some conditions. The proof is based on elementary methods and Cohn’s ones concerning the Diophantine equation x2 + C = yn. Mathematics Subject Classification: 11D61
متن کاملOn a Conjecture on Exponential Diophantine Equations
We deal with a conjecture of Terai (1994) asserting that if a, b, c are fixed coprime integers with min(a, b, c) > 1 such that a+b = c for a certain odd integer r > 1, then the equation a + b = c has only one solution in positive integers with min(x, y, z) > 1. Co-operation man-machine is needed for the proof.
متن کاملThe Exponential Diophantine Equation 2x + by = cz
Let b and c be fixed coprime odd positive integers with min{b, c} > 1. In this paper, a classification of all positive integer solutions (x, y, z) of the equation 2 (x) + b (y) = c (z) is given. Further, by an elementary approach, we prove that if c = b + 2, then the equation has only the positive integer solution (x, y, z) = (1,1, 1), except for (b, x, y, z) = (89,13,1, 2) and (2 (r) - 1, r + ...
متن کاملDiophantine Approximation of the Exponential Function and Sondow’s Conjecture
We begin by examining a hitherto unexamined partial manuscript by Ramanujan on the diophantine approximation of e published with his lost notebook. This diophantine approximation is then used to study the problem of how often the partial Taylor series sums of e coincide with the convergents of the (simple) continued fraction of e. We then develop a p-adic analysis of the denominators of the con...
متن کاملOn the Exponential Diophantine Equation ( 4 m 2 + 1
Let m be a positive integer. Then we show that the exponential Diophantine equation (4m2 + 1)x + (5m2 − 1)y = (3m)z has only the positive integer solution (x, y, z) = (1, 1, 2) under some conditions. The proof is based on elementary methods and Baker’s method. Mathematics Subject Classification: 11D61
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 2003
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa106-4-2