# Why is $n=\frac{2p^2+2pq+2pr+q^2+2qr+r^2}{p+q+r}$, where $n$ is $\text{prime}$, of form such that $p\pm a,p\pm b,$ are $\text{prime}, 1<a<b<n$

Why is $n= \left\lfloor \frac{2p^2+2pq+2pr+q^2+2qr+r^2}{p+q+r} \right\rfloor$, where $n$ is $\text{prime}$, of form such that $p\pm a,p\pm b,$ are $\text{prime}, 1<a<b<n$?

Consider this:

Let $p,q,r$ be $\mathbb{prime}$ where $r<q<p$.

Let $x\pm y$ denote $(x+y\quad\mathbb{and}\quad x-y)$

I conjecture that if the following conditions are true; $$\mathbb{isprime}( n),\quad n = \left\lfloor \frac{2 p^2+2 p q+2 p r+q^2+2 q r+r^2}{p+q+r} \right\rfloor$$ Then $n$ is a prime of such form that $n \pm a$, and $n \pm b$ are prime numbers where $a$ and $b$ are distinct positive integers with $a < b < n$.$\quad$ (A137669(n) where $n>2$)

However, I have no clue as to why this is the case.

Example:

$x = 29, y = 17,z = 11, a = 12, b = 18$

$$71 = \left\lfloor \frac{2 \cdot 29^2+2 \cdot 29 \cdot 17+2 \cdot 29 \cdot 11+17^2+2 \cdot 17 \cdot 11+11^2}{29+17+11} \right\rfloor \,\text{is prime}$$

$$71\pm 12 = \{83,59\} = \{71+12,71-12\},\,\text{are both prime}$$

$$71\pm 18 = \{89,53\} = \{71+18,71-18\},\,\text{are both prime}$$

Series representation:

$$\sum_{n = -\infty}^{\infty}=\left\lfloor\left ( \begin{cases} 1, & n = 1 \\ y+z, & n = 0 \\ {y+z\over(-y-z)^n}, & n\geq2 \end{cases} \right ) x^n \right\rfloor = \left\lfloor \frac{2 p^2+2 p q+2 p r+q^2+2 q r+r^2}{p+q+r} \right\rfloor$$

If anyone can explain why this is the case, it would be very much appreciated.

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I'm afraid your fraction is $$\frac{4090}{57} \approx 71.754$$ –  Will Jagy May 16 '13 at 2:17
@WillJagy: Please note that it is within $\left\lfloor \right\rfloor$, where $\left\lfloor x \right\rfloor = \text{floor}(x)$. I am aware that it is incoherent with the title, but $\LaTeX$ is character count expensive. –  JohnWO May 16 '13 at 2:37

## 1 Answer

Actually, you can represent any odd integer $n$ this way. With your original letters, the fraction is $$\frac{p^2 + (p+q+r)^2}{p+q+r}.$$ You can take $$p = \frac{n-1}{2}, \; \; q+r = \lceil \sqrt n \rceil$$

=====================

   n           p         q + r
3           1           2
5           2           3
7           3           3
9           4           3

11           5           4
13           6           4
15           7           4
17           8           5
19           9           5

21          10           5
23          11           5
25          12           5
27          13           6
29          14           6

31          15           6
33          16           6
35          17           6
37          18           7
39          19           7

41          20           7
43          21           7
45          22           7
47          23           7
49          24           7

51          25           8
53          26           8
55          27           8
57          28           8
59          29           8

61          30           8
63          31           8
65          32           9
67          33           9
69          34           9

71          35           9
73          36           9
75          37           9
77          38           9
79          39           9

81          40           9
83          41          10
85          42          10
87          43          10
89          44          10

91          45          10
93          46          10
95          47          10
97          48          10
99          49          10

101          50          11
103          51          11
105          52          11
107          53          11
109          54          11

111          55          11
113          56          11
115          57          11
117          58          11
119          59          11

121          60          11
123          61          12
125          62          12
127          63          12
129          64          12

131          65          12
133          66          12
135          67          12
137          68          12
139          69          12

141          70          12
143          71          12
145          72          13
147          73          13
149          74          13

151          75          13
153          76          13
155          77          13
157          78          13
159          79          13

161          80          13
163          81          13
165          82          13
167          83          13
169          84          13

171          85          14
173          86          14
175          87          14
177          88          14
179          89          14

181          90          14
183          91          14
185          92          14
187          93          14
189          94          14

191          95          14
193          96          14
195          97          14
197          98          15
199          99          15

201         100          15
203         101          15
205         102          15
207         103          15
209         104          15

211         105          15
213         106          15
215         107          15
217         108          15
219         109          15

221         110          15
223         111          15
225         112          15
227         113          16
229         114          16

231         115          16
233         116          16
235         117          16
237         118          16
239         119          16

241         120          16
243         121          16
245         122          16
247         123          16
249         124          16

251         125          16
253         126          16
255         127          16
257         128          17
259         129          17

261         130          17
263         131          17
265         132          17
267         133          17
269         134          17

271         135          17
273         136          17
275         137          17
277         138          17
279         139          17

281         140          17
283         141          17
285         142          17
287         143          17
289         144          17

291         145          18
293         146          18
295         147          18
297         148          18
299         149          18

301         150          18
303         151          18
305         152          18
307         153          18
309         154          18

311         155          18
313         156          18
315         157          18
317         158          18
319         159          18

321         160          18
323         161          18
325         162          19
327         163          19
329         164          19

331         165          19
333         166          19
335         167          19
337         168          19
339         169          19

341         170          19
343         171          19
345         172          19
347         173          19
349         174          19

351         175          19
353         176          19
355         177          19
357         178          19
359         179          19

361         180          19
363         181          20
365         182          20
367         183          20
369         184          20

371         185          20
373         186          20
375         187          20
377         188          20
379         189          20

381         190          20
383         191          20
385         192          20
387         193          20
389         194          20

391         195          20
393         196          20
395         197          20
397         198          20
399         199          20

401         200          21
403         201          21


====================

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Thanks for showing interest, and for sharing your observation; However, it does not answer my question. –  JohnWO May 16 '13 at 3:40
@JohnWO, you asked why the floor of a certain fraction is always a prime with a special property. The truth is, your expression is, for example, any odd number. In particular, 19 is not one of those special primes, but is easily represented by your fraction. –  Will Jagy May 16 '13 at 3:48
@JohnWO, that is, the thing you conjecture is false. –  Will Jagy May 16 '13 at 3:50
I specify that one should let $p,q,r$ be $\mathbb{prime}$ where $r<q<p$; Your example does not do that. I do not see how this disproves the conjecture. –  JohnWO May 16 '13 at 4:04
@JohnWO, I can't say I noticed the part about $p,q,r$ being prime. Well, I will leave this here regardless. I think the answer with the extra condition is that all primes over, say, 20, are in the list, and your fraction still represents lots of numbers with the restriction. I just picked the simplest pattern. Good luck. –  Will Jagy May 16 '13 at 4:11