My forecast for Mersenne primes as of
June 25, 2007
exp((-1.50474-0.0158 x) x) (1.18878*10^-8exp(3.00948 x)-2.09887*10^7exp(0.0316 x^2) Cos[1.12068 (-81.5706+x) (-13.6123+x)])
{interpolation , mp}
{{250.509, 4423}, {589.969, 9689}, {1346.18, 9941}, {2976.09,  11213}, {6374.66, 19937}, {13229.3, 21701},
{26600.2, 23209}, {51820.5, 44497}, {97810.8, 86243}, {178871., 110503}}
{{316930., 132049}, {544069., 216091}, {904926.397873}, {1.45828*10^6, 859433}, {2.27687*10^6, 1257787},
{3.44431*10^6, 1398269}, {5.04819*10^6, 2976221}, {7.16865*10^6, 3021377}, {9.86298*10^6, 6972593},
{1.31476*10^7, 13466917}}
{{1.31476*10^7, 13466917}, {1.69807*10^7, 20996011}, {2.12486*10^7, 24036583}, {2.57617*10^7,
25964951}, {3.02613*10^7,  30402457}, {3.44405*10^7, 32582657}}
This Model predicts that the 45th mp is around
37,976,944.