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[10001st Prime Number](https://chatgpt.com/c/67998217-17d0-800b-9965-2f34b6e02840)
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**You said:**
What is the 10001st prime number?
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**ChatGPT said:**
Thought about finding prime numbers for a second
The 10,001st prime number is **104,743**. This is a well-known result, for instance, from Project Euler’s Problem 7, which specifically asks for the 10,001st prime.
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**You said:**
What is the 10001st prime number? Use mathematical technique. Don't use programming techniqe.
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**ChatGPT said:**
Thought about prime number calculation for 6 seconds
Below is an outline of how one can determine the 10,001st prime number **using mathematical (theoretical) techniques**, rather than writing or running code. Of course, doing all the details by pure hand is lengthy, but the method itself relies on well-established number-theoretic steps rather than programming.
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## 1. Estimate an Upper Bound for the 10,001st Prime
A common starting point is to use an approximation for the nn-th prime, provided by the Prime Number Theorem. It tells us that for large nn:
pn≈nln(n).pn≈nln(n).
For n=10,001n=10,001:
1. ln(10,001)≈9.2103ln(10,001)≈9.2103 (using known logarithm tables or approximate expansions).
2. So a rough estimate of the 10,001st prime is10,001×9.2103≈92,120.10,001×9.2103≈92,120.
To be safe (because this is only an approximation), we might allow some margin and pick a slightly larger number, say around 120,000 or 130,000, as an upper bound within which we will find the 10,001st prime.
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## 2. Prepare to Check Which Numbers Are Prime
There are two classical, purely mathematical techniques often used to identify all primes up to some bound:
1. **Trial Division by Known Primes**
- You systematically check each candidate number kk by testing divisibility by primes not exceeding kk.
- This approach can be done by hand (theoretically) but is tedious for large numbers because you must keep track of each prime you have found so far.
2. **Sieve of Eratosthenes** (in principle, a mathematical method)
- List all integers up to the chosen upper bound (say 120,000).
- Repeatedly mark off multiples of each known prime starting from 2, 3, 5, etc.
- Those that remain unmarked are prime.
- You don’t have to program this; it can be done with pen and paper on a large grid (as was done historically), though it still involves a lot of manual labor if done entirely by hand.
Either of these approaches is a “mathematical technique,” but in practice, it’s more convenient to do partial calculations, skip multiples systematically, and so on. Nevertheless, no computer code is necessary in principle.
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## 3. Count the Primes Until You Reach the 10,001st
Following either technique:
1. Start from the smallest prime, 2.
2. Keep listing primes in ascending order: 2, 3, 5, 7, 11, 13, and so on.
3. Continue until you have identified exactly 10,001 primes.
4. Record the largest one you encountered in this enumeration.
This method is the concept behind a (non-programmatic) “Sieve” or “Trial Division by Hand” for all primes under an appropriate limit. Historically, mathematicians did precisely these steps (though for smaller ranges) using written tables.
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## 4. Result
By a systematic count (the same count you would get if you carried out an Eratosthenes sieve by hand up to around 120,000), you find:
$104,743$
is the **10,001st prime**.
Though completing the entire sieve or trial division by hand is obviously large-scale manual work, the principle stands purely on classical, paper-and-pencil mathematics, without relying on programming. The number $104,743$ is a well-known and frequently verified result in mathematical circles (famously it appears in Project Euler Problem 7).
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