clc

digits 100

syms p1 p2 p3 t t2 c

c=sym(2^16-16)

range=[1,6] 

for tr=1:10 

    r1=randi(range);

    p1=nthprime(r1);

    r2=randi(range);

    p2=nthprime(r2);

    r3=randi(range);

    p3=nthprime(r3);

    t= sym((p1^sym(12)+p2^sym(12)+p3^sym(12)-sym(3)));

    t2= simplifyFraction(sym(t/c));

    fprintf('%d %d %d\n',p1,p2,p3)

    disp(t)

    disp(t2)

    disp(" ")

end

shows what happens if primes under 16 are used:

Each grouping consists of:

Line 1: the prime values

Line 2: (P1)^12+(P2)^12+(P3)^12-3

Line 3: ((P1)^12+(P2)^12+(P3)^12-3)/(2^16-16)

        ,fraction (in reduced form).

        

In all the random instances shown, the quotient is indeed not an integer.        

c =

65520

range =

     1     6

2 3 7

13841822735

212951119/1008

 

13 13 2

46596170249055

147924349997/208

 

13 13 5

46596414385584

46226601573/65

 

11 2 5

3138672521439

3832322981/80

 

13 11 13

49734598621680

9867975917/13

 

7 2 13

23311926413775

518042809195/1456

 

13 11 11

29574941875920

5868044023/13

 

11 5 7

3152513804544

1684035152/35

 

3 11 2

3138428912255

6897645961/144

 

3 2 2

539630

593/72

When only primes larger than 16 are used, the quotient is always found to be an integer, thus finding no counterexamples--the division always results in an integer:

c =

65520

range =

     7    20

61 53 31

3146395541338100862720

48021909971582736

 

71 17 41

16432246813563414550080

250797417789429404

 

37 59 61

4440129344542958910480

67767541888628799

 

23 41 41

45148895225164392480

689085702459774

 

53 53 59

2761715226752985026160

42150720799038233

 

29 53 53

982871623296657778320

15001093151658391

 

23 19 41

22587618239717272560

344743868127553

 

71 31 47

16526661886533548262240

252238429281647562

 

29 59 29

1779905047805943654960

27165827957966173

 

43 53 67

8713937439686845875600

132996603169823655

 

 

c =

65520

range =

     8   200

1217 167 109

10555697569321349819939295388619816400

161106495258262359889183385052195

 

1097 541 1021

4321114530668068986013220539512701760

65951076475397878296905075389388

 

971 193 257

702474901158201433133028405975876240

10721533900460949834142680188887

 

61 691 929

425077380653636013798482177497201440

6487750009976129636728970963022

 

577 41 1217

10557059357647546111795029813618276960

161127279573375245906517549047898

 

569 359 179

1156304672386028302498250534452080

17648117710409467376346925129

 

719 653 881

243729131475594102482107552838099760

3719919589065844054977221502413

 

157 383 149

9963232927648115006480360204160

152063994622223977510384008

 

1021 367 283

1283249237548421350203970132206552960

19585611073693854551342645485448

 

23 953 563

562210457740920547888838183244264240

8580745692016491878645271417037 

A larger number of random tests with primes above 16 also shows no counterexamples.