Before trying each problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Part A: Make a table of n⁴+1 and 17*2ⁿ-1. The odd prime values of each are bold.

n    n4+1    17*2n-1
0       1        16
1       2        33
2      17        67
3      82       135
4     257       271
5     626       543
6    1297      1087
7    2402      2175
8    4097      4351
9    6562      8703
10  10001     17407
11  14642     34815
12  20737     69631
13  28562    139263
14  38417    278527
15  50626    557055
16  65537   1114111
17  83522   2228223
18 104977   4456447
19 130322   8912895
20 160001  17825791
21 194482  35651583
22 234257  71303167

Part B: Same but with new formulas.

n   21*2n-1  7*4n+1
0       20            8
1       41           29
2       83          113
3      167          449
4      335         1793
5      671         7169
6     1343        28763
7     2687       114689
8     5375       458763
9    10751      1835009
10   21503      7340033
11   43007     29360129
12   86015    117440513 
13  172031    469762049
14  344063   1879048193
15  688127   7516192769
16 1376255  30064771073
17 2752511 120259084289