Is 39 moves the longest a chess game can go moving only pawns?

Solution 1:

Edit: This is an improved version of my previous 39-move solution (using Ross Millikan's suggestion to let the pawns advance as far as possible befrore being captured).

1. c3 b6
2. c4 b5
3. c5 b4
4. c6 b3
5. g3 e6
6. g4 e5
7. g5 e4
8. g6 e3
9. axb3 dxc6
10. fxe3 hxg6
11. h3 a6
12. h4 a5
13. h5 a4
14. h6 a3
15. h7 a2
16. hxg8N axb1N
17. b4 g5
18. b5 g4
19. b6 g3
20. b7 g2
21. bxa8N gxh1N
22. b3 g6
23. b4 g5
24. b5 g4
25. b6 g3
26. b7 g2
27. bxc8N gxf1N
28. e4 c5
29. e5 c4
30. e6 c3
31. e7 c2
32. exf8N cxd1N
33. e3 c6
34. e4 c5
35. e5 c4
36. e6 c3
37. e7 c2
38. d3 f6
39. d4 f5
40. d5 f4
41. d6 f3
42. exd8N f2+

Solution 2:

You can do better by having pawns move more before being captured.

1-4   b2-b6     g7-g3
5     hg        ab
6-9   g3-g7     b6-b2
10    g7xh8N    b2xa1N
11-15 g2-g7     b7-b2
16    g7xf8N    b2xc1N
17-21 a2-a7     h7-h2
22    a7xb8N    h2xg1N
23-26 e2-e6     c7-c3
27    dc        fe
28-31 c3-c7     e6-e2
32    c7xd8N    e2xd1N
33-37 c2-c7     e7-e2
38    f2        e2xf1N
39-42 f3-f7+    d7-d3