twinPrimesQ[tp_]:=tp[[1]]+2==tp[[2]]&&PrimeQ[tp[[1]]]&&PrimeQ[tp[[2]]];
primesList[p_]:=Module[{out={Prime[3],i}},
    For[i=4,Prime[i]<=p,i=i+1,
        Append[out,Prime[i]];
    ];
    out
];
testPrime[p_,pl_]:=Module[{out,found=False,twinPrimes,primeFactors,primeFactorsPowers,i},
    twinPrimes={
        {3 5 prod p-4,3 5 prod p-2},
        {3 5 prod p+2,3 5 prod p+4}
    };
    primeFactors=primesList[p];
    primeFactorsPowers=Tuples[Range[0,pl],primeFactors//Length];
    For[i=1,i<=Length[primeFactorsPowers],i=i+1,
        out=twinPrimes/.prod->Product[primeFactors[[k]]^primeFactorsPowers[[i]][[k]],{k,1,primeFactors//Length}];
        found=twinPrimeQ[out[[1]]]||twinPrimeQ[out[[2]]];
        If[found,Break[]];
    ];
    If[found,out~Select~(twinPrimesQ[#]&),False]//First
];

out=List[]; For[j=4,j<=7,j=j+1,out=out~Append~{j,Prime[j],testPrime[Prime[j],1]}]; out//TableForm