4
$\begingroup$

Last night I asked

Do monic irreducible polynomials assume at least one prime value?

which allowed for a case I had forgotten.

The case of interest is $x^4 + a,$ for $a > 0.$ It is not difficult to show that this is irreducible when $a > 0,$ $a \neq 4 k^4.$ Find all positive integers $m$, such that $n^4+m$ is not prime for any positive integer $n$

That's the question, does irreducible $x^4 + a$ assume at least one prime value?

computer run for $1 \leq a \leq 325$

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

 a 1  x 1 prime  2
 a 2  x 1 prime  3
 a 3  x 2 prime  19
 a 4  x 1 prime  5
 a 5  x 6 prime  1301
 a 6  x 1 prime  7
 a 7  x 2 prime  23
 a 8  x 3 prime  89
 a 9  x 10 prime  10009
 a 10  x 1 prime  11
 a 11  x 6 prime  1307
 a 12  x 1 prime  13
 a 13  x 2 prime  29
 a 14  x 165 prime  741200639
 a 15  x 2 prime  31
 a 16  x 1 prime  17
 a 17  x 12 prime  20753
 a 18  x 1 prime  19
 a 19  x 20 prime  160019
 a 20  x 3 prime  101
 a 21  x 2 prime  37
 a 22  x 1 prime  23
 a 23  x 6 prime  1319
 a 24  x 35 prime  1500649
 a 25  x 2 prime  41
 a 26  x 3 prime  107
 a 27  x 2 prime  43
 a 28  x 1 prime  29
 a 29  x 90 prime  65610029
 a 30  x 1 prime  31
 a 31  x 2 prime  47
 a 32  x 3 prime  113
 a 33  x 8 prime  4129
 a 34  x 5 prime  659
 a 35  x 12 prime  20771
 a 36  x 1 prime  37
 a 37  x 2 prime  53
 a 38  x 9 prime  6599
 a 39  x 10 prime  10039
 a 40  x 1 prime  41
 a 41  x 60 prime  12960041
 a 42  x 1 prime  43
 a 43  x 2 prime  59
 a 44  x 75 prime  31640669
 a 45  x 2 prime  61
 a 46  x 1 prime  47
 a 47  x 18 prime  105023
 a 48  x 5 prime  673
 a 49  x 20 prime  160049
 a 50  x 3 prime  131
 a 51  x 2 prime  67
 a 52  x 1 prime  53
 a 53  x 12 prime  20789
 a 54  x 85 prime  52200679
 a 55  x 2 prime  71
 a 56  x 3 prime  137
 a 57  x 2 prime  73
 a 58  x 1 prime  59
 a 59  x 30 prime  810059
 a 60  x 1 prime  61
 a 61  x 4 prime  317
 a 62  x 21 prime  194543
 a 63  x 2 prime  79
 a   64 =  2^6  all composite  
 a 65  x 6 prime  1361
 a 66  x 1 prime  67
 a 67  x 2 prime  83
 a 68  x 3 prime  149
 a 69  x 10 prime  10069
 a 70  x 1 prime  71
 a 71  x 6 prime  1367
 a 72  x 1 prime  73
 a 73  x 2 prime  89
 a 74  x 255 prime  4228250699
 a 75  x 4 prime  331
 a 76  x 3 prime  157
 a 77  x 6 prime  1373
 a 78  x 1 prime  79
 a 79  x 10 prime  10079
 a 80  x 27 prime  531521
 a 81  x 2 prime  97
 a 82  x 1 prime  83
 a 83  x 72 prime  26873939
 a 84  x 5 prime  709
 a 85  x 2 prime  101
 a 86  x 3 prime  167
 a 87  x 2 prime  103
 a 88  x 1 prime  89
 a 89  x 570 prime  105560010089
 a 90  x 11 prime  14731
 a 91  x 2 prime  107
 a 92  x 3 prime  173
 a 93  x 2 prime  109
 a 94  x 5 prime  719
 a 95  x 18 prime  105071
 a 96  x 1 prime  97
 a 97  x 2 prime  113
 a 98  x 3 prime  179
 a 99  x 10 prime  10099
 a 100  x 1 prime  101
 a 101  x 96 prime  84934757
 a 102  x 1 prime  103
 a 103  x 4 prime  359
 a 104  x 45 prime  4100729
 a 105  x 8 prime  4201
 a 106  x 1 prime  107
 a 107  x 24 prime  331883
 a 108  x 1 prime  109
 a 109  x 30 prime  810109
 a 110  x 3 prime  191
 a 111  x 2 prime  127
 a 112  x 1 prime  113
 a 113  x 6 prime  1409
 a 114  x 5 prime  739
 a 115  x 2 prime  131
 a 116  x 3 prime  197
 a 117  x 4 prime  373
 a 118  x 3 prime  199
 a 119  x 60 prime  12960119
 a 120  x 7 prime  2521
 a 121  x 2 prime  137
 a 122  x 57 prime  10556123
 a 123  x 2 prime  139
 a 124  x 45 prime  4100749
 a 125  x 42 prime  3111821
 a 126  x 1 prime  127
 a 127  x 4 prime  383
 a 128  x 9 prime  6689
 a 129  x 50 prime  6250129
 a 130  x 1 prime  131
 a 131  x 6 prime  1427
 a 132  x 5 prime  757
 a 133  x 2 prime  149
 a 134  x 75 prime  31640759
 a 135  x 2 prime  151
 a 136  x 1 prime  137
 a 137  x 6 prime  1433
 a 138  x 1 prime  139
 a 139  x 10 prime  10139
 a 140  x 9 prime  6701
 a 141  x 2 prime  157
 a 142  x 3 prime  223
 a 143  x 6 prime  1439
 a 144  x 5 prime  769
 a 145  x 4 prime  401
 a 146  x 3 prime  227
 a 147  x 2 prime  163
 a 148  x 1 prime  149
 a 149  x 30 prime  810149
 a 150  x 1 prime  151
 a 151  x 2 prime  167
 a 152  x 3 prime  233
 a 153  x 4 prime  409
 a 154  x 115 prime  174900779
 a 155  x 6 prime  1451
 a 156  x 1 prime  157
 a 157  x 2 prime  173
 a 158  x 3 prime  239
 a 159  x 10 prime  10159
 a 160  x 3 prime  241
 a 161  x 12 prime  20897
 a 162  x 1 prime  163
 a 163  x 2 prime  179
 a 164  x 15 prime  50789
 a 165  x 2 prime  181
 a 166  x 1 prime  167
 a 167  x 12 prime  20903
 a 168  x 13 prime  28729
 a 169  x 10 prime  10169
 a 170  x 3 prime  251
 a 171  x 16 prime  65707
 a 172  x 1 prime  173
 a 173  x 132 prime  303595949
 a 174  x 35 prime  1500799
 a 175  x 2 prime  191
 a 176  x 3 prime  257
 a 177  x 2 prime  193
 a 178  x 1 prime  179
 a 179  x 720 prime  268738560179
 a 180  x 1 prime  181
 a 181  x 2 prime  197
 a 182  x 3 prime  263
 a 183  x 2 prime  199
 a 184  x 5 prime  809
 a 185  x 6 prime  1481
 a 186  x 5 prime  811
 a 187  x 4 prime  443
 a 188  x 3 prime  269
 a 189  x 290 prime  7072810189
 a 190  x 1 prime  191
 a 191  x 6 prime  1487
 a 192  x 1 prime  193
 a 193  x 4 prime  449
 a 194  x 45 prime  4100819
 a 195  x 2 prime  211
 a 196  x 1 prime  197
 a 197  x 6 prime  1493
 a 198  x 1 prime  199
 a 199  x 50 prime  6250199
 a 200  x 3 prime  281
 a 201  x 4 prime  457
 a 202  x 3 prime  283
 a 203  x 6 prime  1499
 a 204  x 5 prime  829
 a 205  x 4 prime  461
 a 206  x 21 prime  194687
 a 207  x 2 prime  223
 a 208  x 7 prime  2609
 a 209  x 30 prime  810209
 a 210  x 1 prime  211
 a 211  x 2 prime  227
 a 212  x 3 prime  293
 a 213  x 2 prime  229
 a 214  x 5 prime  839
 a 215  x 6 prime  1511
 a 216  x 7 prime  2617
 a 217  x 2 prime  233
 a 218  x 9 prime  6779
 a 219  x 40 prime  2560219
 a 220  x 7 prime  2621
 a 221  x 24 prime  331997
 a 222  x 1 prime  223
 a 223  x 2 prime  239
 a 224  x 15 prime  50849
 a 225  x 2 prime  241
 a 226  x 1 prime  227
 a 227  x 6 prime  1523
 a 228  x 1 prime  229
 a 229  x 130 prime  285610229
 a 230  x 3 prime  311
 a 231  x 4 prime  487
 a 232  x 1 prime  233
 a 233  x 24 prime  332009
 a 234  x 5 prime  859
 a 235  x 2 prime  251
 a 236  x 3 prime  317
 a 237  x 14 prime  38653
 a 238  x 1 prime  239
 a 239  x 30 prime  810239
 a 240  x 1 prime  241
 a 241  x 2 prime  257
 a 242  x 9 prime  6803
 a 243  x 4 prime  499
 a 244  x 25 prime  390869
 a 245  x 12 prime  20981
 a 246  x 7 prime  2647
 a 247  x 2 prime  263
 a 248  x 15 prime  50873
 a 249  x 50 prime  6250249
 a 250  x 1 prime  251
 a 251  x 18 prime  105227
 a 252  x 5 prime  877
 a 253  x 2 prime  269
 a 254  x 45 prime  4100879
 a 255  x 2 prime  271
 a 256  x 1 prime  257
 a 257  x 6 prime  1553
 a 258  x 5 prime  883
 a 259  x 10 prime  10259
 a 260  x 27 prime  531701
 a 261  x 2 prime  277
 a 262  x 1 prime  263
 a 263  x 6 prime  1559
 a 264  x 25 prime  390889
 a 265  x 2 prime  281
 a 266  x 3 prime  347
 a 267  x 2 prime  283
 a 268  x 1 prime  269
 a 269  x 30 prime  810269
 a 270  x 1 prime  271
 a 271  x 6 prime  1567
 a 272  x 3 prime  353
 a 273  x 10 prime  10273
 a 274  x 35 prime  1500899
 a 275  x 6 prime  1571
 a 276  x 1 prime  277
 a 277  x 2 prime  293
 a 278  x 3 prime  359
 a 279  x 40 prime  2560279
 a 280  x 1 prime  281
 a 281  x 12 prime  21017
 a 282  x 1 prime  283
 a 283  x 6 prime  1579
 a 284  x 15 prime  50909
 a 285  x 4 prime  541
 a 286  x 3 prime  367
 a 287  x 6 prime  1583
 a 288  x 7 prime  2689
 a 289  x 10 prime  10289
 a 290  x 21 prime  194771
 a 291  x 2 prime  307
 a 292  x 1 prime  293
 a 293  x 18 prime  105269
 a 294  x 5 prime  919
 a 295  x 2 prime  311
 a 296  x 9 prime  6857
 a 297  x 2 prime  313
 a 298  x 3 prime  379
 a 299  x 150 prime  506250299
 a 300  x 19 prime  130621
 a 301  x 2 prime  317
 a 302  x 3 prime  383
 a 303  x 10 prime  10303
 a 304  x 5 prime  929
 a 305  x 6 prime  1601
 a 306  x 1 prime  307
 a 307  x 4 prime  563
 a 308  x 3 prime  389
 a 309  x 20 prime  160309
 a 310  x 1 prime  311
 a 311  x 6 prime  1607
 a 312  x 1 prime  313
 a 313  x 4 prime  569
 a 314  x 45 prime  4100939
 a 315  x 2 prime  331
 a 316  x 1 prime  317
 a 317  x 6 prime  1613
 a 318  x 7 prime  2719
 a 319  x 20 prime  160319
 a 320  x 3 prime  401
 a 321  x 2 prime  337
 a 322  x 5 prime  947
 a 323  x 6 prime  1619
 a   324 =  2^2 3^4  all composite  
 a 325  x 6 prime  1621

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

$\endgroup$
3
  • 1
    $\begingroup$ See OEIS sequence A225766. $\endgroup$ – Robert Israel May 29 '17 at 16:35
  • $\begingroup$ some references at oeis.org/A225768 which concerns $x^6 + a$ $\endgroup$ – Will Jagy May 29 '17 at 16:51
  • $\begingroup$ The question is open, but it might be ineresting that it was proven that there are infinite many primes of the form $x^4+y^2$, where $x$ and $y$ are positive integers. $\endgroup$ – Peter May 30 '17 at 8:25
1
$\begingroup$

Bunyakovsky's conjecture would say that it does. But this is still very much an open question.

$\endgroup$
3
  • $\begingroup$ Robert, good. There is no benefit (as in a proof) when dropping from infinitely many primes to just a single prime? $\endgroup$ – Will Jagy May 29 '17 at 16:32
  • $\begingroup$ @WillJagy I don't think it will help, see this answer and this answer $\endgroup$ – i9Fn May 29 '17 at 17:48
  • 2
    $\begingroup$ My impression is that current methods of analytic number theory generally prove that something exists by proving that an infinite number of things exist. So I don't think that this makes things much easier. $\endgroup$ – Robert Israel May 30 '17 at 1:20

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