
theorem Th39:
  2503 is prime
proof
  733*733=2503*214+1647;
  then
A1: 733*733 mod 2503=1647 by NAT_D:def 2;
A2: 4334=2503*1+1831;
  458*458=2503*83+2015;
  then
A3: 458* 458 mod 2503=2015 by NAT_D:def 2;
A4: 359*64=2503*9+449;
  359*359=2503*51+1228;
  then
A5: 359*359 mod 2503=1228 by NAT_D:def 2;
A6: 1178*712=2503*335+231;
  2015*2015=2503*1622+359;
  then
A7: 2015*2015 mod 2503=359 by NAT_D:def 2;
A8: 1228*449=2503*220+712;
  1228*1228=2503*602+1178;
  then
A9: 1228*1228 mod 2503=1178 by NAT_D:def 2;
A10: 1647*231=2503*152+1;
  1178*1178=2503*554+1022;
  then
A11: 1178*1178 mod 2503=1022 by NAT_D:def 2;
A12: 2503-'1=2503-1 by XREAL_1:233
    .=2502;
  1022*1022=2503*417+733;
  then
A13: 1022*1022 mod 2503=733 by NAT_D:def 2;
A14: 2|^18-'1=2|^18-1 by PREPOWER:11,XREAL_1:233;
  256*256 = 2503*26+458;
  then
A15: 256*256 mod 2503 = 458 by NAT_D:def 2;
A16: 2503-1=139*18 & 139=139|^1;
A17: 2|^1 mod 2503 = 2 by NAT_D:24;
A18: 2|^2 mod 2503=2|^(2*1) mod 2503 .=2*2 mod 2503 by A17,Th11
    .= 4 by NAT_D:24;
A19: 2|^4 mod 2503=2|^(2*2) mod 2503 .=4*4 mod 2503 by A18,Th11
    .= 16 by NAT_D:24;
A20: 2|^8 mod 2503=2|^(2*4) mod 2503 .=16*16 mod 2503 by A19,Th11
    .=256 by NAT_D:24;
A21: 2|^16 mod 2503=2|^(2*8) mod 2503 .= 458 by A20,A15,Th11;
A22: 2|^32 mod 2503=2|^(2*16) mod 2503 .=2015 by A21,A3,Th11;
A23: 2|^64 mod 2503=2|^(2*32) mod 2503 .= 359 by A22,A7,Th11;
A24: 2|^18 mod 2503 = 2|^(16+2) mod 2503 .= 2|^16*2|^2 mod 2503 by NEWTON:8
    .= (458*4) mod 2503 by A18,A21,NAT_D:67
    .= 1832 by NAT_D:24;
A25: 2|^128 mod 2503=2|^(2*64) mod 2503 .=1228 by A23,A5,Th11;
A26: 2|^256 mod 2503=2|^(2*128) mod 2503 .=1178 by A25,A9,Th11;
  2502=139*18+0;
  then
A27: (2|^((2503-'1)div 139)-'1)gcd 2503 = (2|^18 -' 1) gcd 2503 by A12,
NAT_D:def 1
    .= (2|^18-'1+2503*1) gcd 2503 by EULER_1:8
    .= 2503 gcd ((2|^18+2502) mod 2503) by A14,NAT_D:28
    .= 2503 gcd ((1832+(2502 mod 2503)) mod 2503) by A24,NAT_D:66
    .= 2503 gcd ((1832+2502) mod 2503) by NAT_D:24
    .= (1831*1+672) gcd 1831 by A2,NAT_D:def 2
    .= 672 gcd (672*2+487) by EULER_1:8
    .= (487*1+185) gcd 487 by EULER_1:8
    .= 185 gcd (185*2+117) by EULER_1:8
    .= (117*1+68) gcd 117 by EULER_1:8
    .= 68 gcd (68*1+49) by EULER_1:8
    .= (49*1+19) gcd 49 by EULER_1:8
    .= 19 gcd (19*2+11) by EULER_1:8
    .= (11*1+8) gcd 11 by EULER_1:8
    .= 8 gcd (8*1+3) by EULER_1:8
    .= (3*2+2) gcd 3 by EULER_1:8
    .= 2 gcd (2*1+1) by EULER_1:8
    .= 2 gcd 1 by EULER_1:8
    .= 1 by NEWTON:51;
A28: 2|^512 mod 2503=2|^(2*256) mod 2503 .=1022 by A26,A11,Th11;
A29: 2|^1024 mod 2503=2|^(2*512) mod 2503 .=733 by A28,A13,Th11;
A30: 2|^2048 mod 2503=2|^(2*1024) mod 2503 .=1647 by A29,A1,Th11;
  2|^(2503-'1) mod 2503 = 2|^(2048+454) mod 2503 by A12
    .= 2|^2048*2|^454 mod 2503 by NEWTON:8
    .= ((2|^2048 mod 2503)*(2|^(256+198) mod 2503) ) mod 2503 by NAT_D:67
    .= (1647*((2|^256)*(2|^198) mod 2503)) mod 2503 by A30,NEWTON:8
    .= (1647*((1178*(2|^(128+70) mod 2503))mod 2503)) mod 2503 by A26,NAT_D:67
    .= (1647*((1178*((2|^128*2|^70)mod 2503))mod 2503))mod 2503 by NEWTON:8
    .= (1647*((1178*((1228*(2|^(64+6) mod 2503))mod 2503))mod 2503)) mod
  2503 by A25,NAT_D:67
    .= (1647*((1178*((1228*((2|^64*2|^6)mod 2503))mod 2503))mod 2503)) mod
  2503 by NEWTON:8
    .= (1647*((1178*((1228*((359*(2|^(4+2) mod 2503)) mod 2503))mod 2503))
  mod 2503)) mod 2503 by A23,NAT_D:67
    .= (1647*((1178*((1228*((359*((2|^4*2|^2) mod 2503)) mod 2503)) mod 2503
  ))mod 2503)) mod 2503 by NEWTON:8
    .= (1647*((1178*((1228*((359*((16*4) mod 2503)) mod 2503)) mod 2503))mod
  2503)) mod 2503 by A18,A19,NAT_D:67
    .=(1647*((1178*((1228*((359*64) mod 2503))mod 2503))mod 2503)) mod 2503
  by NAT_D:24
    .= (1647*((1178*((1228*449)mod 2503))mod 2503))mod 2503 by A4,NAT_D:def 2
    .= (1647*((1178*712)mod 2503))mod 2503 by A8,NAT_D:def 2
    .= 1647*231 mod 2503 by A6,NAT_D:def 2
    .= 1 by A10,NAT_D:def 2;
  hence thesis by A16,A27,Th25,Th34;
end;
