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Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random from for $m\ll N$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$

Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random where $m\ll N$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$

Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random from the first $2^{128}$ primes for $m= 2^{40}$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$

Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random from for $m\ll N$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$

Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p'q,q'$ be a few odd primes chosen uniformly at random from the first $2^{128}$ primes for $m= 2^{40}$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$

Suppose, $N=p\cdot q$ is the product of two safe primes $p=2p'+1$ and $q=2q'+1$ for some odd primes $p'$ and $q'$.

Let, $p_0,p_1,\ldots,p_m\ll p',q'$ be a few odd primes chosen uniformly at random from the first $2^{128}$ primes for $m= 2^{40}$. My question is that is it possible to have the following congruence at all,

$$p_0^{p_1 \times p_2 \times \cdots\times p_m}\equiv \pm 1 \pmod N?$$