Brahmagupta Fibonacci 身份
哎哎哎::1230【https://www . geeksforgeeks . org/brahmagupta-Fibonacci-identity/
婆罗门格普塔斐波那契恒等式表示两个数的乘积，每个数都是 2 个平方的和，可以用 2 种不同的形式表示为 2 个平方的和。 数学上，
如果 a = p^2 + q^2，b = r^2 + s^2 那么 a * b 可以写成两种不同的形式: =(p^2+q^2)*(r^2+s^2) =(pr–QS)2+(PS+QR)2 ……( 1) =(pr+QS)2+(PS–QR)2 ………( 2)
一些例子是:
a = 5(= 1^2+2^2) b = 25(= 3^2+4^2) ab = 125 a * b 表示为 2 个正方形之和: 2^2+11^2 = 125 5^2+10^2 = 125 em>解释:t8】a = 5 和 b = 25 每个都可以表示为 2 个正方形之和，它们的积 a * b 为 125 可以表示为两个不同形式的 2 个正方形之和。这是根据 的梵天斐波那契恒等式，满足恒等式条件。 a = 13(= 2^2+3^2) b = 41(= 4^2+5^2) ab = 533 将 a * b 表示为 2 个正方形之和: 2^2+23^2 = 533 7^2+22^2 = 533 a = 85(= 6 2+7 2) b = 41(= 4 2+5 2) a * b = 3485 将 a * b 表示为
下面是一个程序，用于验证给定的两个数字的梵天斐波那契恒等式，这两个数字是两个正方形的和。
C++
// CPP code to verify
// Brahmagupta Fibonacci identity
#include <bits/stdc++.h>
using namespace std;
void find_sum_of_two_squares(int a,
int b)
{
int ab = a*b;
// represent the product
// as sum of 2 squares
for (int i = 0; i * i <= ab; i++)
{
for (int j = i; i * i +
j * j <= ab; j++)
{
// check identity criteria
if (i * i + j * j == ab)
cout << i << "^2 + " << j
<< "^2 = " << ab << "\n";
}
}
}
// Driver code
int main()
{
// 1^2 + 2^2
int a = 1 * 1 + 2 * 2;
// 3^2 + 4^2
int b = 3 * 3 + 4 * 4;
cout << "Representation of a * b as sum"
" of 2 squares:\n";
// express product of sum of 2 squares
// as sum of (sum of 2 squares)
find_sum_of_two_squares(a, b);
}
Java 语言(一种计算机语言，尤用于创建网站)
// Java code to verify Brahmagupta
// Fibonacci identity
class GFG
{
static void find_sum_of_two_squares(int a,
int b)
{
int ab = a * b;
// represent the product
// as sum of 2 squares
for (int i = 0; i * i <= ab; i++)
{
for (int j = i; i * i +
j * j <= ab; j++)
{
// check identity criteria
if (i * i + j * j == ab)
System.out.println(i + "^2 + " +
j +"^2 = " + ab);
}
}
}
// Driver code
public static void main(String[] args)
{
// 1^2 + 2^2
int a = 1 * 1 + 2 * 2;
// 3^2 + 4^2
int b = 3 * 3 + 4 * 4;
System.out.println("Representation of a * b " +
"as sum of 2 squares:");
// express product of sum
// of 2 squares as sum of
// (sum of 2 squares)
find_sum_of_two_squares(a, b);
}
}
// This code is contributed
// by Smitha Dinesh Semwal
Python 3
# Python 3 code to verify
# Brahmagupta Fibonacci identity
def find_sum_of_two_squares(a, b):
ab = a * b
# represent the product
# as sum of 2 squares
i=0;
while(i * i <= ab):
j = i
while(i * i + j * j <= ab):
# check identity criteria
if (i * i + j * j == ab):
print(i,"^2 + ",j,"^2 = ",ab)
j += 1
i += 1
# Driver code
a = 1 * 1 + 2 * 2 # 1^2 + 2^2
b = 3 * 3 + 4 * 4 # 3^2 + 4^2
print("Representation of a * b as sum"
" of 2 squares:")
# express product of sum of 2 squares
# as sum of (sum of 2 squares)
find_sum_of_two_squares(a, b)
# This code is contributed by
# Smitha Dinesh Semwal
C
// C# code to verify Brahmagupta
// Fibonacci identity
using System;
class GFG
{
static void find_sum_of_two_squares(int a,
int b)
{
int ab = a * b;
// represent the product
// as sum of 2 squares
for (int i = 0; i * i <= ab; i++)
{
for (int j = i; i * i +
j * j <= ab; j++)
{
// check identity criteria
if (i * i + j * j == ab)
Console.Write(i + "^2 + " + j +
"^2 = " + ab + "\n");
}
}
}
// Driver code
public static void Main()
{
// 1^2 + 2^2
int a = 1 * 1 + 2 * 2;
// 3^2 + 4^2
int b = 3 * 3 + 4 * 4;
Console.Write("Representation of a * b " +
"as sum of 2 squares:\n");
// express product of sum of
// 2 squares as sum of (sum of
// 2 squares)
find_sum_of_two_squares(a, b);
}
}
// This code is contributed
// by Smitha Dinesh Semwal
服务器端编程语言（Professional Hypertext Preprocessor 的缩写）
<?php
// PHP code to verify
// Brahmagupta Fibonacci identity
function find_sum_of_two_squares($a, $b)
{
$ab = $a * $b;
// represent the product
// as sum of 2 squares
for ($i = 0; $i * $i <= $ab; $i++)
{
for ($j = $i; $i * $i +
$j * $j <= $ab; $j++)
{
// check identity criteria
if ($i * $i + $j * $j == $ab)
echo $i ,"^2 + ", $j ,
"^2 = " , $ab ,"\n";
}
}
}
// Driver code
// 1^2 + 2^2
$a = 1 * 1 + 2 * 2;
// 3^2 + 4^2
$b = 3 * 3 + 4 * 4;
echo "Representation of a * b ".
"as sum of 2 squares:\n";
// express product of sum of
// 2 squares as sum of (sum
// of 2 squares)
find_sum_of_two_squares($a, $b);
// This code is contributed by aj_36
?>
java 描述语言
<script>
// JavaScript program to verify Brahmagupta
// Fibonacci identity
function find_sum_of_two_squares(a, b)
{
let ab = a * b;
// represent the product
// as sum of 2 squares
for (let i = 0; i * i <= ab; i++)
{
for (let j = i; i * i +
j * j <= ab; j++)
{
// check identity criteria
if (i * i + j * j == ab)
document.write(i + "^2 + " +
j +"^2 = " + ab + "<br/>");
}
}
}
// Driver code
// 1^2 + 2^2
let a = 1 * 1 + 2 * 2;
// 3^2 + 4^2
let b = 3 * 3 + 4 * 4;
document.write("Representation of a * b " +
"as sum of 2 squares:" + "<br/>");
// express product of sum
// of 2 squares as sum of
// (sum of 2 squares)
find_sum_of_two_squares(a, b);
// This code is contributed by code_hunt.
</script>
Output :
Representation of a * b as sum of 2 squares:
2^2 + 11^2 = 125
5^2 + 10^2 = 125
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