Primes and spirals

In class we looked at how the the polynomial n2 + n + 41 is prime whenever we plugged in values for n.

In what seemed unrelated at the beginning, we also made spirals of numbers, where we would start with one number in the center of the grid, and then "spiral out" counting up by one.
17 18 19 20 21
16 5 6 7 22
15 4 1 8 23
14 3 2 9 24
13 12 11 10 25


They relate to each other when we color all the prime numbers yellow and notice to the yellow line across the diagonal of the graph below (starting at 41). Looking at a bigger table we can see that this streak does in fact stop eventually. The important thing to note here is that just because something is true for small values, doesn't mean it'll be true for everything.

Problem 1 Find a whole number n so that when you plug it into the above equation, the resulting number is not prime. (Hint, you want to try something big, but not so big it's hard to tell whether or not its prime.)

Problem 2 Notice that there are several other streaks (e.g. 53 to 293 in the lower left). Can you find a polynomial that gives one of those lines when you plug in consecutive values of n?

The tables were created with SAGE. You can try making your own online here (to run, press shift-enter in the box).

365366367368369370371372373374375376377378379380381382383
364297298299300301302303304305306307308309310311312313384
363296237238239240241242243244245246247248249250251314385
362295236185186187188189190191192193194195196197252315386
361294235184141142143144145146147148149150151198253316387
360293234183140105106107108109110111112113152199254317388
35929223318213910477787980818283114153200255318389
35829123218113810376575859606184115154201256319390
35729023118013710275564546476285116155202257320391
35628923017913610174554441486386117156203258321392
35528822917813510073544342496487118157204259322393
3542872281771349972535251506588119158205260323394
3532862271761339871706968676689120159206261324395
3522852261751329796959493929190121160207262325396
351284225174131130129128127126125124123122161208263326397
350283224173172171170169168167166165164163162209264327398
349282223222221220219218217216215214213212211210265328399
348281280279278277276275274273272271270269268267266329400
347346345344343342341340339338337336335334333332331330401