Exploring Pythagorean Triplets with a Base Length of 51

Pythagorean triplets are sets of three positive integers (a, b, c) that satisfy the equation a^2 b^2 c^2. When the sides of a right-angled triangle are coprime and the base has a length of 51, determining the possible Pythagorean triplets involves a systematic approach. Let's delve into the process step-by-step.

Step-by-Step Solution

To find the number of Pythagorean triplets with a base length of 51 where the sides are coprime, we can use the properties of generating Pythagorean triplets. First, we need to express 51 in terms of two integers m and n such that a m^2 - n^2 and b 2mn, and then calculate c m^2 n^2.

Step 1: Identify the Base

Given that 51 is an odd number, it can only be expressed as m^2 - n^2. Therefore, we have the equation:

m^2 - n^2 51

This can be factored as:

(m - n)(m n) 51

Step 2: Factor 51

The positive factor pairs of 51 are:

1 times 51 3 times 17

Step 3: Solve for m and n

Let's solve for m and n using each factor pair.

Factor Pair: 1 times 51

From the equations:

m - n 1 m n 51

Add these two equations:

2m 52

Therefore:

m 26

Subtract the first equation from the second:

2n 50

Therefore:

n 25

Here, m 26 and n 25. However, these values are not coprime, as they share the factor 1.

Factor Pair: 3 times 17

From the equations:

m - n 3 m n 17

Add these two equations:

2m 20

Therefore:

m 10

Subtract the first equation from the second:

2n 14

Therefore:

n 7

Here, m 10 and n 7. These values are coprime, as they do not share any common factors other than 1.

Step 4: Generate the Triplet

Now using m 10 and n 7 to generate the triplet:

a m^2 - n^2 10^2 - 7^2 100 - 49 51 b 2mn 2 cdot 10 cdot 7 140 c m^2 n^2 10^2 7^2 100 49 149

The triplet is 51, 140, 149.

Conclusion

Thus, there is one Pythagorean triplet 51, 140, 149 where the sides are coprime.

This method can be applied to similar problems where the base is an odd number. If the base is an even number, a different approach can be used, as shown in the example provided, where the base is 51 (an odd number).

Additional Insights

For a right-angled triangle where one side is 51 and it is not the hypotenuse, the other side can be determined by the formula:

If the number is odd: Other side n^2 - 1/2, n^2 1/2 If the number is even: Other side n^2 - 4/4, n^2 4/4

For 51, an odd number:

The other side is 51^2 - 1/2 26 cdot 50 1300 The other side is 51^2 1/2 1301

The greatest common divisor (GCD) of 51 and 1300 is 1, indicating that the triplet is coprime.

Thus, the triplet is 51, 1300, 1301.