List the common Pythagorean numbers Thank you, there is a list of common Pythagorean numbers?

i=3 j=4 k=5

i=5 j=12 k=13

i=6 j=8 k=10

i=7 j=24 k=25

i=8 j=15 k=17

i=9 j=12 k=15

i=9 j=40 k=41

i=10 j=24 k=26

i=11 j=60 k=61

i=12 j=16 k=20

i=12 j=35 k=37

i=13 j=84 k=85

i=14 j=48 k=50

i=15 j=20 k=25

i=15 j=36 k=39

i=16 j=30 k=34

i=16 j=63 k=65

i=18 j=24 k=30

i=18 j=80 k=82

i=20 j=21 k=29

i=20 j=48 k=52

i=21 j=28 k=35

i=21 j=72 k=75

i=24 j=32 k=40

i=24 j=45 k=51

i=24 j=70 k=74

i=25 j=60 k=65

i=27 j=36 k=45

i=28 j=45 k=53

i=30 j=40 k=50

i=30 j=72 k=78

i=32 j=60 k=68

i=33 j=44 k=55

i=33 j=56 k=65

i=35 j=84 k=91

i=36 j=48 k=60

i=36 j=77 k=85

i=39 j=52 k=65

i=39 j=80 k=89

i=40 j=42 k=58

i=40 j=75 k=85

i=42 j=56 k=70

i=45 j=60 k=75

i=48 j=55 k=73

i=48 j=64 k=80

i=51 j=68 k=85

i=54 j=72 k=90

i=57 j=76 k=95

i=60 j=63 k=87

i=65 j=72 k=97

Common general formulas:

3n, 4n, 5n (n is a positive integer).

2n 1, 2n 2 2n, 2n 2 2n 1 (n is a positive integer).

2 2*(n 1), [2(n 1)] 2 1, [2(n 1)] 2 1 (n is a positive integer).

4) m2 n 2,2mn, m 2 n 2 (m and n are both positive integers, m>n).

2n+1，2n²+2n ，2n²+2n+1

To see whether a group of numbers is a Pythagorean number, first remove the greatest common divisor, and then see if the two larger numbers are 1 apart, and the sum of the two larger numbers is the square of the smallest number.

For example: 39,252,255, first remove the greatest common divisor 3, it becomes 13,84,85, and then look at the difference between the two larger numbers 84,85 by 1, and the sum of 84,85 is 169 is exactly the square of the minimum number 13, so 39,252,255 is a set of Pythagorean numbers.

The Pythagorean number is also known as the Pythagorean number. The Pythagorean number is a set of positive integers that can form the three sides of a right triangle. Pythagorean theorem: The sum of the squares of two right-angled sides A and B of a right-angled triangle is equal to the square of the hypotenuse c (a + b Huaiyu = c).

The Pythagorean number generally refers to three positive integers (a, b, c) that can form the three sides of a right triangle.

i.e. a 2 + b 2 = c 2, a, b, c n

And because the new array (na, nb, nc) obtained by multiplying three numbers in any Pythagorean array (a, b, c) by an integer n at the same time is still the Pythagorean number, we generally want to find a Pythagorean array with a, b, and c buried in each other.

There are two more common and practical ways to use such arrays:

1. When a is an odd number 2n+1 greater than 1, b=2*n 2+2*n, c=2*n 2+2*n+1.

In fact, it is to split the square number of Sun Ranma a into two consecutive natural numbers, for example:

n=1(a,b,c)=(3,4,5).

(a, b, c) = (5, 12, 13) when n = 2, (a, b, c) = (7, 24, 25) when n = 3

The commonly used Pythagorean numbers are and so on.

The Pythagorean number is also known as the Pythagorean number. The Pythagorean number is a set of positive integers that can form the three sides of a right triangle. The Pythagorean number is based on the Pythagorean theorem. The Pythagorean theorem is one of the important mathematical theorems discovered and proven by mankind in the early days.

The Pythagorean theorem states that the sum of the squares of the lengths of the two right-angled sides of a right-angled triangle on a plane (known as hook length and bridge strand length) is equal to the square of the hypotenuse length (known as chord length in ancient times). Conversely, if the sum of the squares of the two sides of a triangle on a plane is equal to the square of the length of the third side, then it is a right triangle (the side opposite the right angle is the third side).

According to the "Zhou Ji Sutra", in the dialogue between Zhou Gong and Shang Gao on numbers in more than 1,000 BC, Shang Gao explained in detail the elements of the Pythagorean theorem with three specific numbers of three, four, five and three as examples.

Ancient Egyptian papyri from 2600 BC had (3,4,5) Pythagorean numbers, and the largest Pythagorean array of ancient Babylonian tablets involved was (12,709,13,500,18,541).

1. When a is an odd number 2n+1 greater than 1, b=2n +2n, c=2n +2n+1.

In effect, it is to split the square number of a into two consecutive natural numbers, for example:

n=1(a,b,c)=(3,4,5).

n=2(a,b,c)=(5,12,13)n=3(a,b,c)=(7,24,25)[1].

Since two successive natural numbers are necessarily coprime, all Pythagorean arrays obtained by this routine are coprime.

2. When a is an even number 2n greater than 4, b=n -1, c=n +1 is to subtract 1 and add 1 to half of the square of a, for example:

n=3(a,b,c)=(6,8,10).

n=4(a,b,c)=(8,15,17)n=5(a,b,c)=(10,24,26)n=6(a,b,c)=(12,35,37).

If you want to determine whether three numbers are Pythagorean numbers, you can only use the Pythagorean theorem, and the sum of the squares of the two smaller numbers in the three numbers is equal to the square of the largest number, which is the Pythagorean number.

In a right-angled triangle, if a and b represent two right-angled edges, and c represents hypotenuse, the Pythagorean theorem can be expressed as a2+b2=c2

The positive integers a, b, and c satisfying this equation are called a set of Pythagorean numbers.

For example, each group can satisfy a2+b2=c2, so they are all Pythagorean arrays (where is the simplest set of Pythagorean numbers).Obviously, if the sides of a right triangle are positive integers, then these three numbers form a set of Pythagorean numbers; Conversely, each set of Pythagorean numbers determines a right triangle with a positive side length of an integer. Therefore, mastering the method of determining the Pythagorean array is of great significance for the study of right triangles.

1 Take any two positive integers m, n, so that 2mn is a perfectly squared number, then.

c=2+9+6=17.

is a set of Pythagorean numbers.

Proof: a, b, c form a set of Pythagorean numbers.

2 Take any two positive integers m, n, (m n), then.

A=M2-N2, B=2Mn, C=M2+N2 form a set of Pythagorean numbers.

For example, when m=4, n=3, a=42-32=7, b=2 4 3=24, c=42+32=25

is a set of Pythagorean numbers.

Proof: a2+b2=(m2-n2)+(2mn)2

m4-2m2n2+n4+4m2n2

m4+2m2n2+4n2

m2+n2)2

C2a, B, and C form a set of Pythagorean numbers.

3 If one of the numbers in the Pythagorean array has been determined, the other two numbers can be determined as follows.

Start by observing whether a known number is odd or even.

1) If it is an odd number greater than 1, it is squared and split into two adjacent integers, then the odd number and these two integers form a set of Pythagorean numbers.

For example, 9 is a number in the Pythagorean number, then is a set of Pythagorean numbers.

Proof: Let an odd number greater than 1 be 2n+1, then square it and split it into two adjacent integers.

2) If it is an even number greater than 2, divide it by 2 and square it, and then subtract 1 from this square number, and add 1 to get two integers and this even number to form a group of Pythagorean numbers.

For example, 8 is a number in the Pythagorean array.

Then, 17 is a set of Pythagorean numbers.

Proof: Let the even number 2n greater than 2, then divide the even number by 2 and then square it, and then subtract the square number by 1 respectively, and add 1 to get the two integers n2-1 and n2+1

2n)2+(n2-1)2=4n2+n4-2n2+1

n4+2n2+1

n2+1)2

2n, n2-1, n2+1 make up a set of Pythagorean numbers.