A Number Theory Problem for 2-12-07: Mission Impossible?

PLS READ THE COMMENTS TO SEE HOW THE LESSON FARED OVER TWO DAYS. DO YOU THINK I HAD TO MODIFY IT A LITTLE OR A LOT TO MAINTAIN THEIR INTEREST?

Thanks to jd2718 for motivating me to include the following famous number-theory conundrum. You can find many references to it on the web but I remember it from the classic text by Hardy and Wright. Give this as a challenge bonus problem or as an activity for your middle or high schoolers. Revise and modify it to make it appropriate for your students. I've tried to turn this into something students can at least attempt, but you could probably make it much more user-friendly...
Remember: My goal is to provide meaningful activities for ALL of your students, not only for honors or accelerated classes. The challenge is to modify them for students who struggle in math. The real test comes when I implement today's problem in my 'skills' class to which I frequently refer. That is my plan -- I'll let you know if it worked or was a disaster! Pls note that these activities also provide practice for those open-ended types of questions that now frequently appear on standardized and state assessments.
Finally, I hope you enjoy these challenges, but my primary target audience is students. I am really interested in reading students' reactions to these.

MISSION IMPOSSIBLE?
Now, boys and girls, we know that, despite all attempts by the most famous mathematicians, no one has yet devised a formula for primes. At one time, it was thought that 2^p-1 would always be prime if p is prime:
2^2-1 = 3; 2^3-1 = 7; 2^5-1 = 31; 2^7-1 = 127 -- all primes. But alas, 2^11-1 - 2047 = (23)(89). Imagine the disappointment!
But I think I found a method that will produce primes EVERY time! If you can do all parts below and prove I'm wrong, you get 5 bonus points. You'll get at least one point for doing part (a).

Ok, I decided to start with my favorite prime 41, since that's the age I once was!
41 + 2 = 43
43 + 4 = 47
47 + 6 = 53
53 + 8 = 61
61 + 10 = 71
71 + 12 = 83
...
Here's your mission:
(a) I listed the first 6 and they're all prime. You can see the pattern, right? Keep the sequence going until you reach 40 numbers. Check that they are all prime by researching a list of primes on the web. So, am I right? Am I famous now?
(b) My method is easy to see but harder to describe algebraically. Here's one way:
To get each number in the list after the first, you can see that I added the next even number to the preceding term. SHOW that the following recursive description produces the first 6 terms:
a(1) = 41; a(n+1) = a(n) + 2n, n = 1,2,3,... where the notation a(n) refers to the nth term. Then explain why this will generate all the terms.
(c) This one is harder but I'll give you a hint: Devise a polynomial that will generate this sequence of numbers when n = 1,2,3,....
Hint: Think of a quadratic like n^2 +.... Just remember, when n is replaced by 1 it has to yield a value of 41!
(d) Show that my amazing sequence 'blows up' when you reach the 41st term. Why? There goes my million dollars from the Clay Institute!
Extension: Is there any other sequence like this? What's special about 41? Happy web-questing...