How do you find the solution to the Pell equation?
solving Pell’s equation and minimizing x satisfies x1 = hi and y1 = ki for some i. This pair is called the fundamental solution. Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell’s equation is found.
What is Pell’s equation used for?
341). The Pell equation was also solved by the Indian mathematician Bhaskara. Pell equations are extremely important in number theory, and arise in the investigation of numbers which are figurate in more than one way, for example, simultaneously square and triangular. .
How do you find the solution of an integer equation?
Let a,b∈Z with a≠0.
- If a divides b, then the equation ax=b has exactly one solution that is an integer.
- If a does not divide b, then the equation ax=b has no solution that is an integer.
Who made Pell’s equation?
(The famous Swiss mathematician Leonhard Euler named the equation after the 1 7 th 17^\text{th} 17th century British mathematician John Pell, to whom he mistakenly attributed a solution method discovered by Pell’s contemporary Lord Brouncker; this name has unfortunately persisted despite evidence of work on the …
What is Chakravala method of algorithms?
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell’s equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE).
Which of the following is the Pell’s equation?
as a solution of Pell’s equation n x 2 + 1 = y 2 nx^{2} + 1 = y^{2} nx2+1=y2. and this is an integer solution to Pell’s equation.
What is the cattle problem?
Archimedes’s cattle problem (or the problema bovinum or problema Archimedis) is a problem in Diophantine analysis, the study of polynomial equations with integer solutions. Attributed to Archimedes, the problem involves computing the number of cattle in a herd of the sun god from a given set of restrictions.
What is the integer of answer?
Answer. An integer, also called a “round number” or “whole number,” is any positive or negative number that does not include decimal parts or fractions. For example, 3, -10, and 1,025 are all integers, but 2.76 (decimal), 1.5 (decimal), and 3 ½ (fraction) are not.
Who introduced cyclic method?
Bhāskara II
The chakravala method (Sanskrit: चक्रवाल विधि) is a cyclic algorithm to solve indeterminate quadratic equations, including Pell’s equation. It is commonly attributed to Bhāskara II, (c. 1114 – 1185 CE) although some attribute it to Jayadeva (c. 950 ~ 1000 CE).
How do you solve a continued fraction?
To calculate a continued fraction representation of a number r, write down the integer part (technically the floor) of r. Subtract this integer part from r. If the difference is 0, stop; otherwise find the reciprocal of the difference and repeat. The procedure will halt if and only if r is rational.
Did Archimedes solve the cattle problem?
Another group attested that Archimedes had no knowledge of triangular numbers and thus solved the problem ignoring this condition. Subject to both these conditions, the smallest solution to the problem was computed by Dr. A. Amthor in 1880 to be 7.766 x 10206544 cattle.
When was the cattle problem of Archimedes solved what is the total number of the cattle of the sun?
Solution. which is a total of 50389082 cattle, and the other solutions are integral multiples of these. Note that the first four numbers are multiples of 4657, a value which will appear repeatedly below. (first solved by Amthor).
How do you solve x3 y3 z3?
x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx).
How do you find the particular solution of a linear Diophantine equation?
Solve the linear Diophantine Equation 20x+16y=500,x,y∈Z+.
- Solution.
- Step 1: gcd(20,16)=4.
- Step 2: A solution is 4125=20(1)(125)+16(−1)(125).
- Step 3: Let u = x – 125 and v = y + 125.
- Step 4: In general, the solution to ax + by = 0 is x=bdk and y=-adk, kZ \ {0}, d=gcd(a,b).
- Step 5: Replace u and v.