for pn=1:15 

  for qn=1:15 

    p=nthprime(pn); q=nthprime(qn);

    lhs=sym(p/(p+1)+(q+1)/q);

    [n,d]=numden(lhs);

    if ceil(lhs)==2 

      disp([p q n d])

      if d==n/2+2

           disp('**')

      end

    end

  end

end

The below lists for p and q that lead to a value whose ceiling is 2, 

p, q, numerator, denominator

where the numerator and denominator are those of the reduced fraction on the right hand side of the equation.

The program points out that p = 2, q = 7 results in a value of 38/21, so that n is 19. In this case q - p = 5.

However, if p=2 and q=5, the fraction 28/15 could be written 56/30 so n would be 28.  I don't know if unreduced fractions are allowed. In this case q-p would be 3.

Take the case p=3, q=5, leading to 39/20. If the fraction were written as 156/80, this would fit n=78, and q-p would be 2.

With p=5, q=7, resulting in 83/42 = 332/168, where n = 166. q-p=2.

These things seem to happen whenever q-p is 2. But we did have that best example when q-p was 5, and that was the only case where the fraction was in its reduced form.

[p, q, num, den]

[2, 3, 2, 1]

[2, 5, 28, 15]

[2, 7, 38, 21]

**

[2, 11, 58, 33]

[2, 13, 68, 39]

[2, 17, 88, 51]

[2, 19, 98, 57]

[2, 23, 118, 69]

[2, 29, 148, 87]

[2, 31, 158, 93]

[2, 37, 188, 111]

[2, 41, 208, 123]

[2, 43, 218, 129]

[2, 47, 238, 141]

[3, 5, 39, 20]

[3, 7, 53, 28]

[3, 11, 81, 44]

[3, 13, 95, 52]

[3, 17, 123, 68]

[3, 19, 137, 76]

[3, 23, 165, 92]

[3, 29, 207, 116]

[3, 31, 221, 124]

[3, 37, 263, 148]

[3, 41, 291, 164]

[3, 43, 305, 172]

[3, 47, 333, 188]

[5, 7, 83, 42]

[5, 11, 127, 66]

[5, 13, 149, 78]

[5, 17, 193, 102]

[5, 19, 215, 114]

[5, 23, 259, 138]

[5, 29, 325, 174]

[5, 31, 347, 186]

[5, 37, 413, 222]

[5, 41, 457, 246]

[5, 43, 479, 258]

[5, 47, 523, 282]

[7, 11, 173, 88]

[7, 13, 203, 104]

[7, 17, 263, 136]

[7, 19, 293, 152]

[7, 23, 353, 184]

[7, 29, 443, 232]

[7, 31, 473, 248]

[7, 37, 563, 296]

[7, 41, 623, 328]

[7, 43, 653, 344]

[7, 47, 713, 376]

[11, 13, 311, 156]

[11, 17, 403, 204]

[11, 19, 449, 228]

[11, 23, 541, 276]

[11, 29, 679, 348]

[11, 31, 725, 372]

[11, 37, 863, 444]

[11, 41, 955, 492]

[11, 43, 1001, 516]

[11, 47, 1093, 564]

[13, 17, 473, 238]

[13, 19, 527, 266]

[13, 23, 635, 322]

[13, 29, 797, 406]

[13, 31, 851, 434]

[13, 37, 1013, 518]

[13, 41, 1121, 574]

[13, 43, 1175, 602]

[13, 47, 1283, 658]

[17, 19, 683, 342]

[17, 23, 823, 414]

[17, 29, 1033, 522]

[17, 31, 1103, 558]

[17, 37, 1313, 666]

[17, 41, 1453, 738]

[17, 43, 1523, 774]

[17, 47, 1663, 846]

[19, 23, 917, 460]

[19, 29, 1151, 580]

[19, 31, 1229, 620]

[19, 37, 1463, 740]

[19, 41, 1619, 820]

[19, 43, 1697, 860]

[19, 47, 1853, 940]

[23, 29, 1387, 696]

[23, 31, 1481, 744]

[23, 37, 1763, 888]

[23, 41, 1951, 984]

[23, 43, 2045, 1032]

[23, 47, 2233, 1128]

[29, 31, 1859, 930]

[29, 37, 2213, 1110]

[29, 41, 2449, 1230]

[29, 43, 2567, 1290]

[29, 47, 2803, 1410]

[31, 37, 2363, 1184]

[31, 41, 2615, 1312]

[31, 43, 2741, 1376]

[31, 47, 2993, 1504]

[37, 41, 3113, 1558]

[37, 43, 3263, 1634]

[37, 47, 3563, 1786]

[41, 43, 3611, 1806]

[41, 47, 3943, 1974]

[43, 47, 4133, 2068]

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