I tried to find a nice expression giving a prime related to the new year $2023$ and tried $$f(n):=2023^n+202^3$$ According to my calculations, this expression is not prime upto $n=13\ 000$.
Is there any reason (small factors , algebraic factors , aurifeuillan factors) preventing this expression to be prime , or do we just have to continue the search and can have hope to finally find a prime ?
The $10^7$-candidates (positive integers for which $f(n)$ has no prime factor below $10^7$) are :
[34, 46, 52, 80, 88, 104, 116, 130, 134, 164, 172, 188, 202, 232, 236, 242, 246, 260, 262, 272, 286, 292, 302, 314, 320, 326, 334, 346, 372, 404, 412, 418, 440, 488, 498, 530, 592, 596, 608, 626, 654, 656, 664, 706, 718, 736, 740, 752, 766, 778, 782, 790, 796, 806, 808, 838, 860, 892, 956, 976, 986, 988, 1004, 1046, 1048, 1082, 1086, 1088, 1102, 1114, 1140, 1142, 1156, 1160, 1166, 1168, 1180, 1208, 1212, 1238, 1254, 1256, 1258, 1270, 1286, 1300, 1310, 1312, 1326, 1336, 1348, 1352, 1354, 1362, 13
78, 1382, 1384, 1394, 1406, 1432, 1436, 1450, 1452, 1466, 1490, 1502, 1504, 1520, 1522, 1532, 1538, 1560, 1586, 1592, 1600, 1606, 1628, 1636, 1646, 1660, 1676, 1690, 1700, 1702, 1712, 1744, 1772, 1796, 1802, 1856, 1868, 1870, 1880, 1922, 1940, 1972, 2006, 2008, 2028, 2038, 2048, 2054, 2062, 2068, 2078, 2110, 2118, 2122, 2132, 2134, 2146, 2152, 2194, 2206, 2210, 2218, 2220, 2222, 2230, 2252, 2264, 2278, 2290, 2302, 2308, 2320, 2348, 2360, 2392, 2398, 2402, 2434, 2440, 2458, 2468, 2470, 2504, 2512
, 2530, 2554, 2570, 2602, 2650, 2684, 2694, 2698, 2726, 2752, 2768, 2780, 2782, 2818, 2822, 2836, 2888, 2902, 2908, 2920, 2936, 2938, 2948, 2978, 2986, 2994, 3000, 3020, 3050, 3064, 3092, 3104, 3116, 3118, 3134, 3140, 3160, 3172, 3182, 3188, 3196, 3200, 3202, 3254, 3296, 3298, 3302, 3310, 3326, 3328, 3338, 3340, 3406, 3410, 3442, 3466, 3468, 3470, 3506, 3520, 3526, 3548, 3566, 3604, 3616, 3646, 3650, 3652, 3692, 3694, 3718, 3730, 3746, 3764, 3770, 3772, 3802, 3826, 3832, 3844, 3856, 3860, 3872,
3910, 3912, 3938, 3968, 3982, 3998, 4010, 4012, 4030, 4040, 4044, 4072, 4078, 4084, 4150, 4162, 4166, 4222, 4226, 4246, 4252, 4268, 4288, 4304, 4316, 4322, 4364, 4376, 4404, 4408, 4418, 4432, 4444, 4448, 4478, 4520, 4526, 4530, 4532, 4582, 4600, 4604, 4616, 4660, 4684, 4688, 4696, 4700, 4722, 4728, 4736, 4738, 4756, 4766, 4778, 4796, 4798, 4808, 4850, 4852, 4864, 4876, 4882, 4890, 4894, 4910, 4922, 4938, 4940, 4948, 4952, 4954, 4966, 4976, 4990, 5002, 5034, 5080, 5090, 5108, 5122, 5128, 5134, 51
46, 5156, 5176, 5188, 5230, 5234, 5248, 5284, 5302, 5312, 5368, 5398, 5402, 5450, 5466, 5470, 5482, 5500, 5522, 5540, 5554, 5566, 5568, 5578, 5584, 5590, 5612, 5620, 5636, 5638, 5650, 5690, 5692, 5708, 5710, 5746, 5772, 5776, 5780, 5786, 5792, 5800, 5806, 5818, 5822, 5834, 5846, 5848, 5876, 5890, 5906, 5926, 5942, 5944, 5960, 6016, 6044, 6046, 6072, 6122, 6142, 6152, 6182, 6184, 6208, 6214, 6222, 6236, 6256, 6262, 6266, 6278, 6308, 6312, 6320, 6332, 6368, 6376, 6382, 6394, 6404, 6416, 6422, 6430
, 6434, 6436, 6446, 6452, 6494, 6502, 6504, 6506, 6530, 6536, 6572, 6592, 6602, 6604, 6614, 6644, 6668, 6688, 6712, 6730, 6742, 6760, 6796, 6802, 6814, 6866, 6870, 6878, 6906, 6922, 6968, 6970, 6994, 7022, 7026, 7034, 7046, 7048, 7060, 7082, 7096, 7102, 7106, 7118, 7130, 7132, 7148, 7158, 7162, 7172, 7202, 7216, 7232, 7244, 7264, 7276, 7286, 7300, 7330, 7368, 7400, 7412, 7418, 7426, 7438, 7460, 7468, 7472, 7484, 7498, 7516, 7522, 7568, 7572, 7606, 7612, 7628, 7638, 7664, 7676, 7726, 7736, 7738,
7752, 7778, 7780, 7820, 7834, 7846, 7850, 7876, 7900, 7906, 7930, 7934, 7978, 7986, 7988, 8012, 8018, 8042, 8048, 8056, 8062, 8068, 8084, 8096, 8110, 8128, 8146, 8168, 8180, 8198, 8212, 8270, 8282, 8320, 8326, 8350, 8356, 8392, 8394, 8434, 8440, 8446, 8452, 8480, 8488, 8508, 8510, 8522, 8578, 8588, 8594, 8600, 8632, 8648, 8660, 8688, 8692, 8698, 8704, 8716, 8800, 8852, 8866, 8870, 8872, 8886, 8888, 8902, 8914, 8926, 8938, 8942, 8950, 8966, 8978, 8984, 8996, 9038, 9042, 9050, 9056, 9062, 9070, 91
12, 9122, 9134, 9146, 9202, 9232, 9236, 9266, 9278, 9280, 9292, 9302, 9304, 9320, 9322, 9350, 9362, 9376, 9392, 9442, 9448, 9460, 9498, 9500, 9512, 9532, 9566, 9568, 9572, 9580, 9596, 9602, 9610, 9614, 9644, 9658, 9668, 9680, 9692, 9712, 9752, 9754, 9766, 9796, 9806, 9826, 9838, 9862, 9888, 9892, 9896, 9904, 9908, 9920, 10016]
A doublecheck or extension of the search range would be appreciated.
Best Answer
Comment: We can narrow the bond for n and only try even numbers:
It can be shown that for every natural number a such as $2023$ there can be infinitely many number like n such that $n| 2023^n+1$. we may write:
$A=2023^n+1 +202^3-1$
$202^3\equiv 1\bmod 3 \Rightarrow 202^3-1\equiv 0\bmod 3=3k$
$2023^n\equiv 1\bmod 3\Rightarrow 2023^n+1\equiv 2\bmod 3=3t+2 $
$(3k, 3t+2)=1$
this means terms $(2023^n+1)$ and $(202^3-1)$ have no common divisors.
Also:
$2023^n+202^3 \equiv 2\bmod 3=3(t+k)+2; t ,even, k , odd$
$2023^2\equiv 1\bmod 3\Rightarrow 2023^{2m}\equiv 2\bmod 3$
that is $n=2m $ must be even.
So we have two restrictions for narrowing the bond:
1- n must be odd
2- n divides $2023^n+1$