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Problem 288

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An enormous factorial

For any prime p the number N(p,q) is defined by N(p,q) = ∑_{n=0 to q} T_n*pⁿ with T_n generated by the following random number generator:

S₀ = 290797
S_n+1 = S_n² mod 50515093
T_n = S_n mod p

Let Nfac(p,q) be the factorial of N(p,q).
Let NF(p,q) be the number of factors p in Nfac(p,q).

You are given that NF(3,10000) mod 3²⁰=624955285.

Find NF(61,10⁷) mod 61¹⁰.

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巨大的阶乘

对于任意素数p，定义N(p,q) = ∑_{n=0 to q} T_n*pⁿ，其中T_n由以下随机数生成器生成：

S₀ = 290797
S_n+1 = S_n² mod 50515093
T_n = S_n mod p

记Nfac(p,q)为N(p,q)的阶乘。
记NF(p,q)为Nfac(p,q)的质因数分解中p的个数。

已知NF(3,10000) mod 3²⁰=624955285。

求NF(61,10⁷) mod 61¹⁰。

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