What is the law of the number of Pythagorean such as 3 4 5

Any set of positive integers that can form the three sides of a right triangle is called the Pythagorean number.

1. Let the length of the three sides of the right triangle be a, b, and c, and the Pythagorean theorem knows a 2 + b 2 = c 2, which is a sufficient and necessary condition for the formation of three sides of a right triangle. Therefore, to require a set of Pythagorean numbers is to solve the indefinite equation x 2 + y 2 = z 2 and find the solution of positive integers.

2. Any even number greater than 2 can constitute a group of Pythagorean numbers, in fact, any odd number greater than 1 2n+1 (n 1) as the edge can also constitute a Pythagorean number, and its three sides are 2n n2+2n, 2n2+2n+1, which can be proved by the inverse theorem of the Pythagorean theorem.

Summary: Observing and analyzing the Pythagorean numbers, it can be seen that they have the following two characteristics:

1. The short right-angled side of a right-angled triangle is an odd number, and the other right-angled edge and the hypotenuse are two consecutive natural numbers.

2. The circumference of a right triangle is equal to the sum of the square of the short right side and the short side itself.

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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

C2 a, b, 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.

Summary of the law of Pythagorean numbers: A positive odd number (except 1) with two consecutive positive integers whose sum is equal to the square of the positive odd number is a set of Pythagorean calendar numbers. Let n be a positive odd number (n≠1), then a set of Pythagorean numbers with n as the minimum value can be:

n、(n²-1)/2、(n²+1)/2。

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

Nature of the Pythagorean number:

1.The number of Pythagorean numbers is divided into two categories, the number of coprime Pythagoreans and the number of non-copied Pythagoreans.

The number of coprime Pythagorean numbers means that a, b, and c have no common factor.

The number of non-mutic Pythagorean is a multiple of the number of mutual Pythagorean.

2.The number of coprime Pythagorean numbers in the format of KitKat split number + even number = odd number

The general formula for coprime Pythagorean numbers is a,b,c= n -m,2nm,n +m,nm are positive integers,n>m,n,m coprime, n+m= odd number.

The formula for the Pythagorean number term is:

a,b,c= 2knm, k(n-m) k(n +m) k,n,m are any positive integers, n>m

There are only two kinds of Pythagorean numbers, odd + even = odd and even + even = even.

The general term formula means that given any set of Pythagorean numbers a, b, and c, the ternary equation can be solved to obtain the unique value of k, n, and m (n, m coprime), and vice versa.

3.The number of coprime Pythagoreans, a can be any odd number (excluding 1), b can be a multiple of any 4, c can be [a multiple of 4 + 1, and is a prime number] and their product.

Number of Pythagoreans. The three rules are: 1. Everything that can form a right triangle.

A set of positive integers on three sides.

This is called the Pythagorean number. 2. In a set of Pythagorean numbers, when the smallest side is odd, its square is exactly equal to the sum of two other consecutive positive integers. 3. In a set of Pythagorean numbers, when the smallest side is even, its square is just equal to twice the sum of two consecutive integers.

Rule 1: In the Pythagorean numbers (3,4,5), (5,12,13), (7,24,25) (9,40,41), we find:

By (3, 4, vertical with 5) there are: 32

By (5, 12, 13) there are: 52

By (7, 24, 25) have: 72

By (9, 40, 41) there is: 92

That is, in a set of Pythagorean numbers, when the smallest side is odd, its square is exactly equal to the sum of two other consecutive positive integers. Therefore, we generalize it to the general and thus derive the following formula:

2n+1）2

4n24n+1=（2n2

2n）+（2n2

2n+1）（2n+1）2

2n22n）2

2n22n+1）2

n is a positive integer).

Pythagorean number formula 1: (2n+1, 2n2.)

2n，2n2

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

Rule 2: In the Pythagorean number (6,8,10), (8,15,Yulu 17), (10,24,26), we find:

By (6, 8, 10) there are: 62

By (8, 15, 17) there are: 82

By (10, 24, 26) there are: 102

That is, in a set of Pythagorean numbers, when the smallest side is an even number, its square is just equal to twice the sum of the two successive integers, and by generalization, another formula can be obtained:

2n）24n2

2[（n21）+（n2

2n）2（n2

n2n 2 and n is a positive integer).

Pythagorean number formula two: (2n, n2

1, n21) (n 2 and n is a positive integer).

Like 3, 4, and 5, it can become three positive integers of the length of the three sides of a right triangle, called the Pythagorean number. So what are the rules of the Pythagorean number? Let's learn about it with me for your reference.

The Pythagorean number, 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 triangle is equal to the square of the hypotenuse c (a + b = c).

The Pythagorean theorem is known in the West as the pythagorean theorem, and it is named after the Greek philosophers and mathematicians of the 6th century BC. It is reasonable to consider him one of the most important fundamental theorems in mathematics because his inferences and generalizations are widely quoted. Despite his name, he is also one of the oldest theorems in ancient civilizations, and was in fact discovered by the ancient Babylonians, who had raised ants more than a thousand years before the Pythagoras, as evidenced by the table of numbers on the Plimpton 322 tablets, which date to about 1700 BC.

There are more than 400 ways to prove the Pythagorean theorem from ancient times to the present.

Rule 1: In a set of Pythagorean numbers, when the smallest side is odd, its square is exactly the sum of two other consecutive positive integers.

Rule 2: In a set of Pythagorean numbers, when the smallest side is even, it squares exactly equal to two consecutive odd numbers, or 2 times the sum of two consecutive even numbers.

Rule 3: In a set of Pythagorean numbers, if the first number is odd, then the other two numbers, one number is half of its square minus 1, and one number is half of its square plus 1.

a=m,b=(m2 k-k) 2,c=(m2k+k) 2 (where m3).

When m is determined to be an odd number of any 3, k=.

When m is determined to be an even number of any 4, k=.

The basic Pythagorean number and the derived Pythagorean number can be found by exactly. For example, when m is determined to be an even number 432, m=432 and 24 different sets of k values are substituted into b=(m 2 k-k) 2 and c=(m 2 k+k) 2; because k==That is, when the right-angled edge A=432, there are 24 different groups of another right-angled edge B and hypotenuse C, and the basic Pythagorean number and the derived Pythagorean number are obtained together. The number of groups of Pythagorean numbers can also be obtained directly by formulas.

The Pythagorean number, also known as the Pythagorean number, is a set of positive integers that can form the three sides of a right triangle. Next, I will share with you 3 rules of the number of Pythagoreans.

1.In a set of Pythagorean numbers, when the smallest side is odd, it squares exactly the sum of two other consecutive positive integers.

2.In a set of Pythagorean numbers, when the smallest side is an even number, its squared row is exactly equal to two consecutive odd numbers, or twice the sum of two consecutive even numbers.

3.In a set of Pythagorean numbers, if the first number is odd, then the other two numbers, one is half of its square minus 1, and one is half of its square plus one.

1.Odd number formula: Split into two consecutive numbers after squared.

5 2 = 25, 25 = 12 + 13, so 5, 12, 13 is a set of Pythagorean numbers.

7 2 = 49, 49 = 24 + 25, so 7, 24, 25 is a set of Pythagorean numbers.

9 2 = 81, 81 = 40 + 41, so 9, 40, 41 is a set of Pythagorean numbers.

2.Even number formula: Half of the square is divided into two numbers with a difference of 2.

8 2 = 64, 64 2 = 32, 32 = 15 + 17, so 8, 15, 17 is a set of Pythagorean numbers.

10 2 = 100, 100 2 = 50, 50 = 24 + 26, so 10, 24, 26 is a set of Pythagorean numbers.

12 2 = 144, 144 2 = 72, 72 = 35 + 37, so 12, 35, 37 is a set of Pythagorean numbers.

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

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

We know that, like 3, 4, 5, there are three positive integers that can become the three sides of a right triangle, called the Pythagorean number. What is the law of the Pythagorean number, let's classify it below

1. The length of the shortest side is an odd number, observe the Pythagorean number in the table below:

According to the ** above, we can find that the above Pythagorean numbers have certain characteristics.

Wherein, a=n+(n+1)=2n+1, b=2n(n+1)=2n2 +2n, c=2n(n+1)+1= 2n2 +2n+1, easy to verify:

2n+1)2+(2n2 +2n)2=(2n2 +2n+1)2, that is, when the length of the shortest side is an odd number, the number of Pythagorean pairs conforms to the above law.

2. When the length of the shortest side is even, observe the number of Pythagorean in the following **:

When the shortest side is an even number, a=2(n+1)=2n+2, b=n2 +2n, c= n2 +2n+2, easy to verify:

2n+2)2+(n2 +2n)2=(n2 +2n+2)2, that is, when the length of the shortest side is even, the number of Pythagorean conforms to the above rule.

1. The origin of the Pythagorean theorem.

The Pythagorean theorem is also called the Shang Gao theorem, and it is called the Pythagorean theorem in the West In ancient China, the shorter right angle side in the right triangle was called the hook, the longer right angle side was called the strand, and the hypotenuse was called the chord As early as more than 3,000 years ago, the Zhou Dynasty mathematician Shang Gao put forward the Pythagorean theorem in the form of "hook three, strand four, string five", and later people further discovered and proved that the trilateral relationship of the right triangle was as follows: the sum of squares of the two right angles is equal to the square of the hypotenuse.

2. The application of the Pythagorean theorem is circumferential.

The Pythagorean theorem reveals the quantitative relationship between the three sides of a right triangle, which is only applicable to right triangles, and does not have this characteristic for the three sides of an acute triangle and an obtuse triangle, so when applying the Pythagorean theorem, it must be clear that the object under investigation is a right triangle.

3. Application of the Pythagorean theorem.

Knowing the length of any two sides of a right triangle, find the third side in.

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then, <>

Knowing one side of a right triangle, we can get the quantitative relationship between the other two sides.

The Pythagorean theorem can be used to solve some practical problems.