Consider all 3-term geometric sequences with first term 1 and with common ratio the square of an integer between 1 and 1000. How many of these 1000 geometric sequences have the property that the sum of the 3 terms is prime?
so we are looking for
1 + x^2 + x^4 being a prime
x^4 + x^2 + 1
= x^4 + 2x^2 + 1 - x^2
= (x^2 + 1)^ - x^2 ---- a difference of squares
= (x^2 + 1 +x)(x^2 + 1 -x)
if x = 1 , we get
(1+1+1)(1) = 3 --- a prime number
for any other value of x, the value of each bracket > 1
and 1+x^2 + x^4 is the product of at least two factors
thus : 1 + 1^2 + 1^4 is the only such case
1 1 1 is the only case
111 is incorrect
@Tsong, the answer is not 111, the answer is 3 :P
sorry, the answer is 1 :P
thanxxxx
To find the number of 3-term geometric sequences with the given conditions, we need to determine the common ratios that satisfy the conditions and then count the number of primes that can be obtained as the sum of the terms.
Let's break down the problem into steps:
Step 1: Find the common ratios
The common ratio must be the square of an integer between 1 and 1000. To find these ratios, we can iterate through all the integers between 1 and 1000 and check if their squares meet the criteria. We can create a set to store the valid ratios to avoid duplicates.
Step 2: Calculate the sum of the terms
Once we have the valid common ratios, we can calculate the sum of the three terms for each geometric sequence. The sum can be found using the formula:
Sum = a + ar + ar^2
Step 3: Check for prime sums
For each calculated sum, we need to check if it is a prime number. We can implement a function that checks the primality of a number, using an efficient algorithm like the Sieve of Eratosthenes or a probabilistic primality test like the Miller-Rabin test.
Step 4: Count the number of sequences that satisfy the conditions
Finally, we count the number of sequences that have a prime sum.
Let's write a program that follows these steps:
```python
from sympy import isprime
def count_prime_sums():
ratios = set()
for i in range(1, 1001):
ratio = i ** 2
if 1 < ratio < 1000:
ratios.add(ratio)
count = 0
for ratio in ratios:
sum_of_terms = 1 + ratio + ratio**2
if isprime(sum_of_terms):
count += 1
return count
print(count_prime_sums())
```
This program uses the `isprime` function from the `sympy` library to check if a number is prime. Make sure to install the `sympy` library before running this program.
After executing the program, you will get the count of 3-term geometric sequences satisfying the given conditions, where the sum of the terms is prime.