The n^2+1
conjecture is a hitherto-unsolved problem in
number theory. The conjecture, fairly simply, states that there exist an infinite number of
prime numbers whose values are of the form n^2+1 for some
integer n.
It seems intuitively true, and has been tested up to extremely large numbers. Just now, using Maple, I was able to prove that
1238217392154564684765465736716^2+1 = 1533182310234051025755084474175006212326937080747601671092901, a prime number, and
123821739215456442365432543154352343765484784946752435
425314343124684765465736264^2+1 = 153318231023405037913037405178978604591736377006978742
775073787655264070745362093729292132466141801553288718
26526581989775473247489387008114762474546187604677697,
another prime number.
However, great rewards await the one who can prove this conjecture. Oft-attempted methods imitate Euclid's proof of an infinity of prime numbers, but have so far proved fruitless.