Some useful conclution for bruteforce to solve number theory problem

Правка en2, от OtterZ, 2024-09-19 09:45:57

Here are some useful conclution for bruteforce to solve number theory problem,I hope you can know something about it and solve number theory problems more easily.

1.The number of factors of a intelliger

First of all,$$$\sum_{i = 1} ^ n \operatorname{d}(n) = \sum_{i = 1} ^ n [\frac{n}{i}] \approx n \ln n$$$.

Then I've found out that the number of factors of a intelliger($$$\operatorname{d}(n)$$$) is usually small,and to make sure,I made a code to get the maxinum number of the number of factors,and get:

  1. For $$$n \le 10 ^ 4,\max \operatorname{d}(n) <= 68$$$;
  2. For $$$n \le 5 \times 10 ^ 4,\max \operatorname{d}(n) <= 100$$$;
  3. For $$$n \le 10 ^ 5,\max \operatorname{d}(n) <= 128$$$;
  4. For $$$n \le 2 \times 10 ^ 5,\max \operatorname{d}(n) <= 160$$$;
  5. For $$$n \le 3 \times 10 ^ 5,\max \operatorname{d}(n) <= 180$$$;
  6. For $$$n \le 5 \times 10 ^ 5,\max \operatorname{d}(n) <= 200$$$;
  7. For $$$n \le 10 ^ 6,\max \operatorname{d}(n) <= 240$$$;
  8. For $$$n \le 5 \times 10 ^ 6,\max \operatorname{d}(n) <= 384$$$;
  9. For $$$n \le 10 ^ 7,\max \operatorname{d}(n) <= 448$$$;

So if your solution of a problem is $$$\operaotrname{O}(n\max \operatorname{d}(a_i))$$$ or $$$\operaotrname{O}(\sum \operatorname{d}(a_i))$$$,it might be correct because for $$$a_i \le 10 ^ 7$$$,it's sure that $$$\operatorname{d}(a_i) \le 500$$$.

examples:

2.Euler's Function: $$$\operaotrname{O}(log_2(v))$$$ times to $$$1$$$.

It's sure that $$$\phi(n) \le \frac{n}{2}$$$ for $$$2 | n$$$,and $$$2 | \phi(n)$$$ for $$$n > 1$$$.So if you use operation $$$x = \phi(x)$$$ for $$$x = n$$$ initially,it will become $$$1$$$ in $$$\operatorname{O}(\log_2 n)$$$ times.

Теги number theory, brute force, maths

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en33 Английский OtterZ 2024-09-30 08:21:37 58 Tiny change: '/D)\n- [20' -> '/D)\n- [2005D](https://codeforces.net/contest/2005/problem/D)\n- [20'
en32 Английский OtterZ 2024-09-27 09:54:33 58
en31 Английский OtterZ 2024-09-27 08:33:12 390
en30 Английский OtterZ 2024-09-22 16:01:05 58
en29 Английский OtterZ 2024-09-22 05:40:38 58
en28 Английский OtterZ 2024-09-21 16:31:51 92
en27 Английский OtterZ 2024-09-21 15:59:13 20
en26 Английский OtterZ 2024-09-21 15:57:49 118
en25 Английский OtterZ 2024-09-21 15:50:08 144
en24 Английский OtterZ 2024-09-21 06:36:44 63
en23 Английский OtterZ 2024-09-21 06:07:56 2 Tiny change: 'conclution for naive' -> 'conclutions for naive'
en22 Английский OtterZ 2024-09-20 11:02:07 117
en21 Английский OtterZ 2024-09-20 10:59:30 18
en20 Английский OtterZ 2024-09-20 10:58:55 437
en19 Английский OtterZ 2024-09-20 10:45:05 12 Tiny change: 'er:Timosh]to remind ' -> 'er:Timosh] to remind '
en18 Английский OtterZ 2024-09-20 10:44:52 10 Tiny change: 'ser:Timosh,2024-9-20]to remind' -> 'ser:Timosh]to remind'
en17 Английский OtterZ 2024-09-20 10:44:29 10 Tiny change: 'ser:Timosh]to remind' -> 'ser:Timosh,2024-9-20]to remind'
en16 Английский OtterZ 2024-09-20 10:37:09 29
en15 Английский OtterZ 2024-09-20 10:35:51 2 Tiny change: 'd}(n) <= 68$;\n2. For' -> 'd}(n) <= 64$;\n2. For'
en14 Английский OtterZ 2024-09-20 10:29:16 397
en13 Английский OtterZ 2024-09-20 10:11:11 2 Tiny change: 'on diverse.\n- [user:' -> 'on diverse;\n- [user:'
en12 Английский OtterZ 2024-09-20 10:10:58 394
en11 Английский OtterZ 2024-09-20 08:45:40 20 Tiny change: 'v)$ primes.This can ' -> 'v)$ primes ($2 ^ k$ the worst).This can '
en10 Английский OtterZ 2024-09-20 08:38:08 1 Tiny change: 'e to thanks:\n\n- [us' -> 'e to thank:\n\n- [us'
en9 Английский OtterZ 2024-09-20 08:37:56 887
en8 Английский OtterZ 2024-09-20 08:20:32 23
en7 Английский OtterZ 2024-09-19 11:52:50 5 Tiny change: '{O}(\log_2(v))$ times t' -> '{O}(\log_2 n)$ times t'
en6 Английский OtterZ 2024-09-19 10:15:54 484 (published)
en5 Английский OtterZ 2024-09-19 09:47:36 2 Tiny change: 'or $\operaotrname{O}(\' -> 'or $\operatorname{O}(\'
en4 Английский OtterZ 2024-09-19 09:47:22 20
en3 Английский OtterZ 2024-09-19 09:46:32 17
en2 Английский OtterZ 2024-09-19 09:45:57 7 Tiny change: 'ox n \ln n\n\nI've found' -> 'ox n \ln n$.\n\nThen I've found'
en1 Английский OtterZ 2024-09-19 09:45:35 1816 Initial revision (saved to drafts)