I discovered a relatively good approximation formula for \(\displaystyle \pi(x), \ \) which gives the exact number of prime numbers
less than or equal to the value of x. For example, \(\displaystyle \ \pi(10) = 4, \ \) because there are four prime numbers less than
or equal to 10. My formula works best in the interval of about \(\displaystyle \ 10 \le x \le 3,000.\)
Approximation formula: \(\displaystyle \ \ Ceiling \ function \ of \ \)\(\displaystyle \ \bigg[ \dfrac{x}{\log_{10}{(x^2)}} \bigg] \)
I cherry-picked multiples of 25 for certain x-values, where each approximation formula's value equals the corresponding \(\displaystyle \pi(x) \) value.
x \(\displaystyle \ \ \ \ \ \ \ \)| Approx. formula value = \(\displaystyle \ \pi(x) \)
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\(\displaystyle \ \ \ \) 25 \(\displaystyle \ \ \ \ \ \ \ \ \ \) 9
\(\displaystyle \ \ \ \) 50 \(\displaystyle \ \ \ \ \ \ \ \) 15
\(\displaystyle \ \) 100 \(\displaystyle \ \ \ \ \ \ \ \) 25
\(\displaystyle \ \) 125 \(\displaystyle \ \ \ \ \ \ \ \) 30
\(\displaystyle \ \) 150 \(\displaystyle \ \ \ \ \ \ \ \) 35
\(\displaystyle \ \) 175 \(\displaystyle \ \ \ \ \ \ \ \) 40
\(\displaystyle \ \) 225 \(\displaystyle \ \ \ \ \ \ \ \) 48
\(\displaystyle \ \) 250 \(\displaystyle \ \ \ \ \ \ \ \) 53
\(\displaystyle \ \) 550 \(\displaystyle \ \ \ \ \ \) 101
\(\displaystyle \ \) 575 \(\displaystyle \ \ \ \ \ \) 105
\(\displaystyle \ \) 850 \(\displaystyle \ \ \ \ \ \) 146
\(\displaystyle \ \) 975 \(\displaystyle \ \ \ \ \ \) 164
1375 \(\displaystyle \ \ \ \ \ \) 220
1475 \(\displaystyle \ \ \ \ \ \) 233
1775 \(\displaystyle \ \ \ \ \ \) 274
1875 \(\displaystyle \ \ \ \ \ \) 287
1900 \(\displaystyle \ \ \ \ \ \) 290
2000 \(\displaystyle \ \ \ \ \ \) 303
2100 \(\displaystyle \ \ \ \ \ \) 317
2175 \(\displaystyle \ \ \ \ \ \) 326
2725 \(\displaystyle \ \ \ \ \ \) 397
2775 \(\displaystyle \ \ \ \ \ \) 403
2800 \(\displaystyle \ \ \ \ \ \) 407
2825 \(\displaystyle \ \ \ \ \ \) 410
2850 \(\displaystyle \ \ \ \ \ \) 413
2900 \(\displaystyle \ \ \ \ \ \) 419
2925 \(\displaystyle \ \ \ \ \ \) 422
2975 \(\displaystyle \ \ \ \ \ \) 429
____________________________________________________
You may wish to experiment with this approximation formula on your own.
less than or equal to the value of x. For example, \(\displaystyle \ \pi(10) = 4, \ \) because there are four prime numbers less than
or equal to 10. My formula works best in the interval of about \(\displaystyle \ 10 \le x \le 3,000.\)
Approximation formula: \(\displaystyle \ \ Ceiling \ function \ of \ \)\(\displaystyle \ \bigg[ \dfrac{x}{\log_{10}{(x^2)}} \bigg] \)
I cherry-picked multiples of 25 for certain x-values, where each approximation formula's value equals the corresponding \(\displaystyle \pi(x) \) value.
x \(\displaystyle \ \ \ \ \ \ \ \)| Approx. formula value = \(\displaystyle \ \pi(x) \)
-------------------------------------------
\(\displaystyle \ \ \ \) 25 \(\displaystyle \ \ \ \ \ \ \ \ \ \) 9
\(\displaystyle \ \ \ \) 50 \(\displaystyle \ \ \ \ \ \ \ \) 15
\(\displaystyle \ \) 100 \(\displaystyle \ \ \ \ \ \ \ \) 25
\(\displaystyle \ \) 125 \(\displaystyle \ \ \ \ \ \ \ \) 30
\(\displaystyle \ \) 150 \(\displaystyle \ \ \ \ \ \ \ \) 35
\(\displaystyle \ \) 175 \(\displaystyle \ \ \ \ \ \ \ \) 40
\(\displaystyle \ \) 225 \(\displaystyle \ \ \ \ \ \ \ \) 48
\(\displaystyle \ \) 250 \(\displaystyle \ \ \ \ \ \ \ \) 53
\(\displaystyle \ \) 550 \(\displaystyle \ \ \ \ \ \) 101
\(\displaystyle \ \) 575 \(\displaystyle \ \ \ \ \ \) 105
\(\displaystyle \ \) 850 \(\displaystyle \ \ \ \ \ \) 146
\(\displaystyle \ \) 975 \(\displaystyle \ \ \ \ \ \) 164
1375 \(\displaystyle \ \ \ \ \ \) 220
1475 \(\displaystyle \ \ \ \ \ \) 233
1775 \(\displaystyle \ \ \ \ \ \) 274
1875 \(\displaystyle \ \ \ \ \ \) 287
1900 \(\displaystyle \ \ \ \ \ \) 290
2000 \(\displaystyle \ \ \ \ \ \) 303
2100 \(\displaystyle \ \ \ \ \ \) 317
2175 \(\displaystyle \ \ \ \ \ \) 326
2725 \(\displaystyle \ \ \ \ \ \) 397
2775 \(\displaystyle \ \ \ \ \ \) 403
2800 \(\displaystyle \ \ \ \ \ \) 407
2825 \(\displaystyle \ \ \ \ \ \) 410
2850 \(\displaystyle \ \ \ \ \ \) 413
2900 \(\displaystyle \ \ \ \ \ \) 419
2925 \(\displaystyle \ \ \ \ \ \) 422
2975 \(\displaystyle \ \ \ \ \ \) 429
____________________________________________________
You may wish to experiment with this approximation formula on your own.
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