Are there prime numbers not irreducible in $\mathcal{O}_{\mathbb{Q}(\sqrt{577})}$?

Here is a theoretical approach, which uses a bit of class field theory to show that there will be infinitely many such primes.

Let $H$ be the Hilbert class field of $K=\mathbb Q(\sqrt{577})$, the maximal, everywhere unramified abelian extension of $K$.

Theorem: A prime ideal $\mathfrak p\subset\mathcal O_K$ is principal if and only if it splits completely in $H$.

You are looking for rational primes $p$ which split in in $\mathbb Q(\sqrt{577})$ and such that the prime ideals above $p$ are principal. Hence, the primes you are after (except maybe for the finitely many ramified primes) are exactly the primes which split completely in $H$. The Cebotarev density theorem guarantees that there will be infinitely many such primes.

This argument generalises to any number field $K$: either it is a UFD or there are infinitely many primes which are norms from $K$.

---

The following Sage code spits out some of the primes you're after ($577$ is missing because it is ramified)

K.<a> = NumberField(x^2 - 577)
L = K.hilbert_class_field('b')

for p in prime_range(15000):    
    splits = false
    for P in L.primes_above(p):
        if P.residue_class_degree() == 1 and P.absolute_ramification_index() == 1: 
            splits = true
            break
    if splits: print p

With output beginning

283 293 433 541 569 719 787 941 1097 1187 1429 1451 1531 1579 1663 1867 2003 2029 

I should probably add that, as $$ 1 + 24^2 = 577 \equiv 1 \pmod 8, $$ it follows that the integral quadratic forms $$ x^2 + 23 xy - 12 y^2 $$ and $$ x^2 - 577 y^2 $$ represent exactly the same ODD numbers. This includes the odd primes in which you are interested.

prime = x^2 - 577 y^2
prime    283  x  74  y  3
prime    293  x  51  y  2
prime    433  x  289  y  12
prime    541  x  433  y  18
prime    569  x  99  y  4
prime    577  x  577  y  24
prime    719  x  36  y  1
prime    787  x  218  y  9
prime    941  x  57  y  2
prime    1097  x  195  y  8

This argument is elementary: if $$ x(x + 23 y) - 12 y^2 $$ is odd, it means both $x$ and $x+23y$ are odd. However, $x$ being odd means $23 y = x + 23 y - x$ is even, therefore $y$ is even. Write $y = 2 w.$ Now we have $$ x^2 + 46 xw - 48 w^2. $$ Next, take $v = x + 23 w,$ so $x = v - 23 w.$ The result is $$ v^2 - 577 w^2. $$

These are the droids you seek:

jagy@phobeusjunior:~$ ./Conway_Positive_Primes 1 23 -12 5000 577
           1          23         -12   original form 

           1          23         -12   Lagrange-Gauss reduced 



 Represented (positive) primes up to  5000

   283   293   433   541   569   577   719   787   941  1097
  1187  1429  1451  1531  1579  1663  1867  2003  2029  2083
  2203  2339  2551  2671  2693  2999  3023  3083  3089  3253
  3257  3271  3593  3607  3643  3779  3877  4021  4127  4253
  4339  4409  4457  4517  4643  4793  4937

==============================================================

I wrote something about how binary quadratic forms map to number field orders: http://math.blogoverflow.com/2014/08/23/binary-quadratic-forms-over-the-rational-integers-and-class-numbers-of-quadratic-%EF%AC%81elds/

For what it is worth, the class number is $7.$ The principal form also represents $-1.$

577    factored    577

    1.             1          23         -12   cycle length             6
    2.             2          23          -6   cycle length             6
    3.             3          23          -4   cycle length             6
    4.             4          23          -3   cycle length             6
    5.             6          23          -2   cycle length             6
    6.             6          19          -9   cycle length            10
    7.             9          19          -6   cycle length            10

  form class number is   7

================================================================

The ordering above is the way my program displays the forms, very handy for some purposes. To see the group operation using Dirichlet's method for composition, better to make all the middle coefficients 33, although this makes all coefficients positive. The forms are still indefinite. I ought to mention that these forms are not in the same order as before; there is a bijection with $SL_2 \mathbb Z$ equivalence.

 1     33    128
 2     33     64
 4     33     32
 8     33     16
16     33      8
32     33      4
64     33      2

================================================================

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle 1 23 -12

  0  form              1          23         -12


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   1 23 -12   delta  -1     ambiguous  
1  form   -12 1 12   delta  1
2  form   12 23 -1   delta  -23
3  form   -1 23 12   delta  1     ambiguous            -1 composed with form zero  
4  form   12 1 -12   delta  -1
5  form   -12 23 1   delta  23
6  form   1 23 -12


  form   1 x^2  + 23 x y  -12 y^2 

minimum was   1rep   x = 1   y = 0 disc 577 dSqrt 24  M_Ratio  576
Automorph, written on right of Gram matrix:  
-49  -1152
-96  -2257
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$

The theoretical answer above is correct (although this answer has been edited to be less informative), but one can also give a concrete example. $$(719)=(36+\sqrt{577})(36-\sqrt{577})$$