## TPTP Problem File: RAL039^1.p

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```%------------------------------------------------------------------------------
% File     : RAL039^1 : TPTP v7.5.0. Released v7.0.0.
% Domain   : Real Algebra (Quadratic equations)
% Problem  : International Mathematical Olympiad, 2003, Problem 2
% Version  : [Mat16] axioms : Especial.
% English  : Find all pairs (m, n) of positive integers such that m^2 / (2 m
%            n^2 - n^3 + 1) is a positive integer.

% Refs     : [Mat16] Matsuzaki (2016), Email to Geoff Sutcliffe
%          : [MI+16] Matsuzaki et al. (2016), Race against the Teens - Benc
% Source   : [Mat16]
% Names    : IMO-2003-2.p [Mat16]

% Status   : Theorem
% Rating   : ? v7.0.0
% Syntax   : Number of formulae    : 3485 (   0 unit;1199 type;   0 defn)
%            Number of atoms       : 45347 (2209 equality;22713 variable)
%            Maximal formula depth :   35 (   9 average)
%            Number of connectives : 39643 ( 104   ~; 233   |;1176   &;36004   @)
%                                         (1095 <=>;1031  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  : 2408 (2408   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1246 (1199   :;   0   =;   0  @=)
%                                         (   0  !!;   0  ??;   0 @@+;   0 @@-)
%            Number of variables   : 8058 (  66 sgn;7085   !; 431   ?; 406   ^)
%                                         (8058   :; 136  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
%            Arithmetic symbols    : 1976 (   6 prd;   9 fun;  23 num;1938 var)
% SPC      : TH1_THM_EQU_ARI

% Comments : Theory: PA; Score: 7; Author: Yiyang Zhan;
%            Generated: 2014-11-19
%            ^ [V_mn_dot_0: ( 'ListOf' @ \$int )] :
%            ? [V_l: \$int] :
%              ( ( \$greater @ V_l @ 0 )
%              & ( ( V_mn_dot_0
%                  = ( 'cons/2' @ \$int @ ( \$product @ 2 @ V_l ) @ ( 'cons/2' @ \$int @ V_l @ ( 'nil/0' @ \$int ) ) ) )
%                | ( V_mn_dot_0
%                  = ( 'cons/2' @ \$int @ V_l @ ( 'cons/2' @ \$int @ ( \$product @ 2 @ V_l ) @ ( 'nil/0' @ \$int ) ) ) )
%                | ( V_mn_dot_0
%                  = ( 'cons/2' @ \$int @ ( \$difference @ ( \$product @ 8 @ ( 'int.^/2' @ V_l @ 4 ) ) @ V_l ) @ ( 'cons/2' @ \$int @ ( \$product @ 2 @ V_l ) @ ( 'nil/0' @ \$int ) ) ) ) ) ) )
%------------------------------------------------------------------------------
include('Axioms/MAT001^0.ax').
%------------------------------------------------------------------------------
thf(p_qustion,conjecture,
( 'find/1' @ ( 'ListOf' @ \$int )
@ ^ [V_mn: ( 'ListOf' @ \$int )] :
? [V_m: \$int,V_n: \$int] :
( ( V_mn
= ( 'cons/2' @ \$int @ V_m @ ( 'cons/2' @ \$int @ V_n @ ( 'nil/0' @ \$int ) ) ) )
& ( \$greater @ V_m @ 0 )
& ( \$greater @ V_n @ 0 )
& ( 'int.is-divisible-by/2' @ ( 'int.^/2' @ V_m @ 2 ) @ ( \$sum @ ( \$product @ 2 @ ( \$product @ V_m @ ( \$product @ V_n @ V_n ) ) ) @ ( \$sum @ ( \$uminus @ ( 'int.^/2' @ V_n @ 3 ) ) @ 1 ) ) )
& ( \$greater @ ( \$sum @ ( \$product @ 2 @ ( \$product @ V_m @ ( \$product @ V_n @ V_n ) ) ) @ ( \$sum @ ( \$uminus @ ( 'int.^/2' @ V_n @ 3 ) ) @ 1 ) ) @ 0 ) ) )).

%------------------------------------------------------------------------------
```