Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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 1

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

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

share|improve this answer
    
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
1  
@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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.