The Frontier Fields program is obtaining deep Hubble and Spitzer Space Telescope images of new “blank” fields and nearby fields gravitationally lensed by massive galaxy clusters. The Hubble images of the lensed fields are revealing nJy sources (AB mag > 31), the faintest galaxies yet observed. In this paper, we present high-redshift (z > 6) number count predictions for the full program and cand...

Suppose a 1 < a 2 < : : : < a Z are distinct integers in f1; : : :; N g. If a 0 +" 1 a 1 +" 2 a 2 + + " Z a Z is prime for all choices of " 1 ; " 2 ; : : : ; " Z 2 f0; 1g, then Z (9=2 + o(1)) log N= log log N. The argument uses a modiied form of Gallagher's Larger Sieve, and shows that if Z > (9=2 + o(1)) log N= log log N , then for some prime p < Z 2 , the subset sums " 1 a 1 + " 2 a 2 + + " Z...

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