> restart;
> P := nextprime(37109437);

                            P := 37109441

> M := 12345; N:=100:

                              M := 12345

> for B from 0 to 100 do 
> facs:=Factor(x^2 - M - B) mod P;
> if degree(op(1,facs),x) = 1
> then A1 := op(2,op(1,facs));
>      A2 := op(2,op(2,facs));
>      if(A1 mod N = 0) then print(M+B,A1); break; fi;
>      if(A2 mod N = 0) then print(M+B,A2); break; fi;
> fi;od;

                facs := (x + 19461865) (x + 17647576)


                                 2
                        facs := x  + 37097095


                                 2
                        facs := x  + 37097094


                                 2
                        facs := x  + 37097093


                                 2
                        facs := x  + 37097092


                                 2
                        facs := x  + 37097091


                 facs := (x + 36122314) (x + 987127)


                                 2
                        facs := x  + 37097089


                facs := (x + 16892339) (x + 20217102)


                                 2
                        facs := x  + 37097087


                facs := (x + 11048008) (x + 26061433)


                facs := (x + 17735244) (x + 19374197)


                                 2
                        facs := x  + 37097084


                facs := (x + 12684875) (x + 24424566)


                                 2
                        facs := x  + 37097082


                facs := (x + 15884182) (x + 21225259)


                                 2
                        facs := x  + 37097080


                facs := (x + 18624244) (x + 18485197)


                                 2
                        facs := x  + 37097078


                facs := (x + 24861547) (x + 12247894)


                facs := (x + 14153493) (x + 22955948)


                                 2
                        facs := x  + 37097075


                facs := (x + 22985620) (x + 14123821)


                                 2
                        facs := x  + 37097073


                                 2
                        facs := x  + 37097072


                 facs := (x + 4247074) (x + 32862367)


                 facs := (x + 4432322) (x + 32677119)


                 facs := (x + 27982482) (x + 9126959)


                 facs := (x + 31401541) (x + 5707900)


                            12373, 5707900

>