Intereting Posts

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)
Non-backtracking closed walks and the Ihara zeta function (Updated with partial attempt)
What is the least value of $k$ for which $B^k = I$?
General solution to wave equation of half-line with nonhomogeneous Neumann boundary
When is $c^4-72b^2c^2+320b^3c-432b^4$ a positive square?
What is the difference between a Hamel basis and a Schauder basis?
For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$?
Proof of a combination identity:$\sum \limits_{j=0}^n{(-1)^j{{n}\choose{j}}\left(1-\frac{j}{n}\right)^n}=\frac{n!}{n^n}$
$\text{Evaluate:} \lim_{b \to 1^+} \int_1^b \frac{dx}{\sqrt{x(x-1)(b-x)}}$
How to show that $\lim \frac{1}{n} \sum_{i=1}^n \frac{1}{i}=0 $?
Learning general relativity
Why is associativity required for groups?
What are cohomology rings good for?
If the entries of a positive semidefinite matrix shrink individually, will the operator norm always decrease?
Topological counterexample: compact, Hausdorff, separable space which is not first-countable

Let $n$ be 113. Use $n+C\log^2n$ to find the next consecutive prime or at least approximately how far away it is. Will you show me how to work this out step by step to show me how to use this formula? What is the constant $C$? What is the reason why $(log(n))^2$ is used in this conjecture?

The gap from 113 to 127 is 14. So $C\log^2$113 should be equal to 14 or slightly greater than 14, but less than 22.

$(ln(113))^2$ is approximately 22.348.

- The square of n+1-th prime is less than the product of the first n primes.
- When is $(p - 1)! + 1$ a power of $p$?
- A sum of fractional parts.
- Find all positive integers $a, b, c$ such that $a^2+1$ and $b^2+1$ are both primes and $(a^2+1)(b^2+1)=c^2+1$
- Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$.
- How many resistors are needed?

Is the conjecture that there is ways a prime between $n$ and $n+C\log^2n$ the same as Cramér’s conjecture $O((\log n)^2)$?

Is this conjecture somehow derived from the prime counting function?

- The modular curve X(N)
- $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$
- Group cohomology or classical approach for class field theory?
- Prove: The positive integers cannot be partitioned into arithmetic sequences (using Complex Analysis)
- What is the ratio of rational to irrational real numbers?
- Simplying linear recurrence sum with binomials
- Proofs that every natural number is a sum of four squares.
- How can I prove my conjecture for the coefficients in $t(x)=\log(1+\exp(x)) $?
- Calculation of product of all coprimes of number less than itself
- $5$-adic expansion of $−2$

$n+C\log^2n$ is *not* a good approximation of the next prime after $n$. It’s much more likely to be around $n+\log n.$ Rather, the *largest* prime gap between any two primes up to $n$ is in the neighborhood of $C\log^2n.$ It’s like the difference between the time it would take to run 26 miles (a day?) vs. the world record for a marathon (~2:03). Similarly, the prime after 10^18 is 10^18 + 3 but the record gap up to 10^18 is 1442.

It’s believed that, for any $C<2/e^\gamma\approx1.229,$ the largest gap up to $x$ will be greater than $C\log^2x$ infinitely often. I don’t know of anyone brave enough to conjecture what would be good enough, but it wouldn’t be surprising if for any $C>2/e^\gamma$ there were only finitely many (even 0?) gaps $q-p$ where $q-p>C\log^2q.$

Cramér’s conjecture is precisely the assertion that there exists some $C$ such that for all large enough $x$ (say, $x\ge3$) there is always a prime between $x$ and $x+C\log^2x$. (That’s what the big-O statement *means*.)

Left this as a link at an earlier question, I think that was deleted; anyway, there is no predicting where the next prime occurs.

Here is a table of extreme behavior. I had the computer read in the numbers, then put in a final column with

$$ \color{magenta}{ \frac{g}{(\log p)^2}} $$

The table below is what people have been able to compute so far, with primes as far apart as possible. See http://en.wikipedia.org/wiki/Prime_gap

The fact that can be worked up from this table and the text at the wikipedia page is that, for prime

$$ 11 \leq p < 4 \cdot 10^{18}, $$ there is **at least one** prime $q$ with

$$ p < q < p + (\log p)^2. $$

The prime number theorem says that the expected value of the very next prime is $p + \log p,$ but sometimes that guess is too small, sometimes too large.

```
Stolen from
http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm
the size of the gap is g
next are the number of decimal digits in p
for 4 * 10^18 > p >= 11, g < log^2 p = (log p)^2.
Oh, logarithms base e == 2.718281828459
==================================
g digits of p p log p g/log p g/log^2 p
1 1 1 2 0.693147 1.4427 2.08137
2 2 1 3 1.09861 1.82048 1.65707
3 4 1 7 1.94591 2.05559 1.05637
4 6 2 23 3.13549 1.91357 0.610294
5 8 2 89 4.48864 1.78228 0.397065
6 14 3 113 4.72739 2.96147 0.626449
7 18 3 523 6.25958 2.87559 0.45939
8 20 3 887 6.78784 2.94644 0.434076
9 22 4 1129 7.02909 3.12985 0.445271
10 34 4 1327 7.19068 4.72835 0.657566
11 36 4 9551 9.1644 3.92824 0.428642
12 44 5 15683 9.66033 4.55471 0.471486
13 52 5 19609 9.88374 5.26116 0.532305
14 72 5 31397 10.3545 6.95352 0.671548
15 86 6 155921 11.9571 7.19238 0.601515
16 96 6 360653 12.7957 7.50254 0.586334
17 112 6 370261 12.822 8.73501 0.681254
18 114 6 492113 13.1065 8.698 0.663642
19 118 7 1349533 14.1153 8.35974 0.592248
20 132 7 1357201 14.1209 9.34782 0.661983
21 148 7 2010733 14.514 10.197 0.702566
22 154 7 4652353 15.3529 10.0307 0.653342
23 180 8 17051707 16.6518 10.8097 0.649161
24 210 8 20831323 16.852 12.4615 0.739466
25 220 8 47326693 17.6726 12.4487 0.704405
26 222 9 122164747 18.6209 11.9221 0.640254
27 234 9 189695659 19.0609 12.2764 0.644062
28 248 9 191912783 19.0726 13.003 0.681764
29 250 9 387096133 19.7742 12.6427 0.639356
30 282 9 436273009 19.8938 14.1753 0.712549
31 288 10 1294268491 20.9812 13.7266 0.654231
32 292 10 1453168141 21.097 13.8408 0.656056
33 320 10 2300942549 21.5566 14.8447 0.688637
34 336 10 3842610773 22.0694 15.2247 0.689855
35 354 10 4302407359 22.1824 15.9586 0.719423
36 382 11 10726904659 23.096 16.5396 0.716125
37 384 11 20678048297 23.7523 16.1668 0.680642
38 394 11 22367084959 23.8309 16.5332 0.693772
39 456 11 25056082087 23.9444 19.0441 0.795349
40 464 11 42652618343 24.4764 18.9571 0.774506
41 468 12 127976334671 25.5751 18.299 0.715502
42 474 12 182226896239 25.9285 18.281 0.705055
43 486 12 241160624143 26.2087 18.5434 0.707529
44 490 12 297501075799 26.4187 18.5475 0.702059
45 500 12 303371455241 26.4382 18.912 0.715328
46 514 12 304599508537 26.4423 19.4386 0.735133
47 516 12 416608695821 26.7554 19.2858 0.720819
48 532 12 461690510011 26.8582 19.8078 0.737495
49 534 12 614487453523 27.1441 19.6728 0.724756
50 540 12 738832927927 27.3283 19.7597 0.723048
51 582 13 1346294310749 27.9284 20.839 0.746159
52 588 13 1408695493609 27.9737 21.0198 0.751412
53 602 13 1968188556461 28.3081 21.266 0.751232
54 652 13 2614941710599 28.5923 22.8034 0.797536
55 674 13 7177162611713 29.6019 22.7688 0.769166
56 716 14 13829048559701 30.2578 23.6633 0.782057
57 766 14 19581334192423 30.6056 25.0281 0.817762
58 778 14 42842283925351 31.3885 24.7861 0.789655
59 804 14 90874329411493 32.1405 25.0152 0.778307
60 806 15 171231342420521 32.774 24.5926 0.750369
61 906 15 218209405436543 33.0165 27.4408 0.831126
62 916 16 1189459969825483 34.7123 26.3884 0.760203
63 924 16 1686994940955803 35.0617 26.3535 0.751632
64 1132 16 1693182318746371 35.0654 32.2825 0.920639
65 1184 17 43841547845541059 38.3194 30.8982 0.806335
66 1198 17 55350776431903243 38.5525 31.0745 0.806032
67 1220 17 80873624627234849 38.9317 31.337 0.804922
68 1224 18 203986478517455989 39.8568 30.7099 0.770506
69 1248 18 218034721194214273 39.9234 31.2598 0.782995
70 1272 18 305405826521087869 40.2604 31.5943 0.784749
71 1328 18 352521223451364323 40.4039 32.8681 0.813489
72 1356 18 401429925999153707 40.5338 33.4536 0.825325
73 1370 18 418032645936712127 40.5743 33.7652 0.832181
74 1442 18 804212830686677669 41.2286 34.9757 0.848335
75 1476 19 1425172824437699411 41.8008 35.3103 0.844728
g digits of p p log p g/log p g/log^2 p
==================================
```

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