# Do irreducible polynomials $x^4 + a$ assume a prime value at least once

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 $1 \leq a \leq 325$

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 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


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• – Robert Israel May 29 '17 at 16:35
• some references at oeis.org/A225768 which concerns $x^6 + a$ – Will Jagy May 29 '17 at 16:51
• 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. – Peter May 30 '17 at 8:25