Find all primes less than 100 that can each be written in all three ways:
p=a^2+b^2=c^2+2 e^2=f^2+3 g^2, and show how it's done with each one. The numbers you're squaring must be integers; show your thinking. thanksi don't get it at all
why does noone answer my question and everyone elses
I'll bite. I am interested in the question, but I need to know more about the "rules".
It sounds like a number theory problem, but you marked it as grade 8. Can you tell me a little more about what you have been learning about with prime numbers?
Do a and b have to be distinct?
For example, does
2=1²+1² count?
Do c,d have to be distinct?
Do e,f have to be distinct?
Did your teacher give an example? If yes, can you post the example?
Ex this is what it says
Some primes can be expressed as the sum of two squares, as in 13=2^2 +3^2; some can't:7=/= a^2+b^2. Other primes can be c^2+2 e^2, like 11 =3^2+2
* 1^2; still others are f^2 + 3 g^2, like 31 =2^2 +3 *3^2 that's what the sheet says he didn't really explain he just said do it and me and my friends are all confused and yes its grade 8. we only learned like what numbers are prime/composite but were doing like lcm &gcf so this question was kinda random
I believe this exercise is to encourage organization of work.
Indeed, there are primes under 100 that satisfy these properties, so your teacher does not make you work for no reason. To answer my earlier question, some integers are re-used in composing the same prime.
The key is to search systematically. It may look like a lot of work, but it is not that difficult.
Count the number of primes between 1 and 100 (1 is not a prime). Make a table of four columns and 25 rows. The first column contains the primes, 2,3,5,7,11...
The second column is the sum a²+b&sups;, the third is the sum c²+2d&sups;, and the fourth is the sum e²+3f&sups;.
Start from 1 and 1:
- calculate 1²+1²=2. It is not a prime, do nothing.
- calculate 1²+2*1²=3. It is a prime, so in the square corresponding to the prime 3 and under (third) column for c & d, mark 1²+2*1².
- calculate 1²+3*1²=4. It is not a prime, do nothing.
Repeat for (1,2), (1,3)....until (1,9). 10 is not a candidate, since 10²=100, the upper limit.
Next, do (2,1), (2,2), ...
then (3,1), (3,2)....
until (9,9).
If a row contains three checks, this is your answer. You should find two primes.
If your teacher shows you a mathematical formula to find the answer, I would like to hear about it.
woah dude how come u know so much
and thank you so so so so so much for your help :) ill let you know
Correction:
The second column is the sum a²+b², the third is the sum c²+2d², and the fourth is the sum e²+3f².
Another advantage of the exercise is that by the end of your work, you will be very familiar with all the primes under one hundred.
You're welcome!
To find all primes less than 100 that can be written in all three ways mentioned (p=a^2+b^2, p=c^2+2e^2, p=f^2+3g^2), we need to check each prime number individually and see if it satisfies these conditions. Let's go through each step to find these primes and demonstrate how it's done with an example.
1. Start by listing all prime numbers less than 100:
- 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
2. For each prime number, we need to find integer pairs (a, b), (c, e), and (f, g) that satisfy the given equations.
Let's take the example of p = 5:
- To find a^2 + b^2 = 5, we can start by trying different combinations of (a, b) with integers. In this case, there are no integer solutions because 5 cannot be written as the sum of two perfect squares.
- To find c^2 + 2e^2 = 5, we need to try different combinations of (c, e). Again, there are no integer solutions for this equation.
- To find f^2 + 3g^2 = 5, we can try different combinations of (f, g), and in this case, there are no integer solutions.
Hence, the prime number 5 does not satisfy all three equations.
3. Continuing this process for each of the prime numbers less than 100, we can determine if they can be written in all three ways.
By checking all the prime numbers in the given range, we find that the following prime numbers satisfy all three conditions:
- 2: (1, 1), (1, 0), (-1, 1)
- 13: (3, 2), (2, -3), (-3, 2)
- 37: (6, 1), (1, -6), (-6, 1)
- 61: (5, 6), (5, -6), (-5, 6)
- 73: (4, 7), (4, -7), (-4, 7)
Therefore, the prime numbers less than 100 that can each be written in all three ways are 2, 13, 37, 61, and 73.