
theorem :: Problem 59
  not ex f being non-empty natural-valued increasing Arithmetic_Progression st
    for i,fi being Nat st fi = f.i holds
      fi is perfect_power
  proof
    assume ex f being non-empty natural-valued increasing
      Arithmetic_Progression st
    for i,fi being Nat st fi = f.i holds
      fi is perfect_power; then
    consider f being non-empty natural-valued increasing
      Arithmetic_Progression such that
A1: for i,fi being Nat st fi = f.i holds
      fi is perfect_power;
AA: dom f = NAT by FUNCT_2:def 1;
    reconsider b = f.0 as Nat;
    f.0 < f.1 by SEQM_3:def 1,AA; then
    f.1 - f.0 > f.0 - f.0 by XREAL_1:14; then
    reconsider a = difference f as Nat by NUMBER06:def 5;
KK: f = ArProg (b,a) by NUMBER06:6;
    consider p being Prime such that
P1: p > a + b by NEWTON:72;
P2: p > b by P1,NAT_1:12;
    reconsider p2 = p^2 as Nat;
    a gcd p = 1
    proof
      assume a gcd p <> 1; then
      not a,p are_coprime; then
      consider d being Nat such that
Z2:   d divides a & d divides p & d <> 1 by PYTHTRIP:def 1;
      d = 1 or d = p by Z2,INT_2:def 4; then
      p <= a by NAT_D:7,Z2;
      hence thesis by P1,NAT_1:12;
    end; then
    a gcd p2 = 1 by WSIERP_1:6; then
    consider x,y being Nat such that
Z1: a * x - p2 * y = 1 by WSIERP_1:31;
oo: p - b > b - b by P2,XREAL_1:14; then
    reconsider k = (p - b) * x as Nat;
S1: a * k + b = a * x * (p - b) + b
             .= (1 + p2 * y) * (p - b) + b by Z1
             .= p2 * y * (p - b) + p;
    reconsider akb = a * k + b as Nat;
FZ: not p2 divides (a * k + b)
    proof
      assume
SZ:   p2 divides (a * k + b);
SA:   p > 1 by INT_2:def 4;
      reconsider pb = p - b as Nat by oo;
      p2 divides p2 * (y * pb);
      hence thesis by SquareNotDiv,SA,SZ,S1,NAT_D:10;
    end;
Z2: akb = f.k by KK,NUMBER06:7;
FC: p2 = p |^2 by NEWTON:81;
    akb is not perfect_power
    proof
      assume
N1:   akb is perfect_power;
      akb = p * (p * y * (p - b) + 1) by S1; then
      p divides akb;
      hence thesis by FZ,FC,KeyPerfectPower,N1;
    end;
    hence thesis by A1,Z2;
  end;
