Three odd primes summing into [x - N0, x - 2] (circle-method core)

For a natural number $x$, assume:

• $87 \cdot 10^{35} \le x$, an inequality between natural numbers (so $x \ge 8{.}7 \times 10^{36}$); and
• $\iota(x) \le e^{3100}$, where $\iota(x)$ is the image of $x$ under the canonical embedding $\mathbb{N} \hookrightarrow \mathbb{R}$ and $e^{3100} = \exp(3100)$ is the real exponential at $3100$ — this is a comparison in $\mathbb{R}$, equivalent to $x \le \lfloor e^{3100} \rfloor$ as naturals.

Under these two hypotheses the statement asserts the existence of a natural number $m$ satisfying all of the following simultaneously:

Both constraints on $m$ are written additively, in natural-number arithmetic, with no truncated subtraction anywhere; together they place $m$ in the closed interval

i.e. $m$ is at most $400{,}000{,}000{,}000{,}000$ below $x$ and is strictly below $x$, missing it by at least $2$ (so $m = x$ and $m = x - 1$ are both excluded, while $m = x-2$ is permitted). Since $x \ge 87\cdot 10^{35}$, this window is non-empty and lies well inside $\mathbb{N}$.

Here "prime" is primality of a natural number (so $p_i \ge 2$ and $p_i$ has no divisor other than $1$ and itself), and "odd" for a natural number $p$ means $p = 2k+1$ for some $k \in \mathbb{N}$; the two conditions are imposed separately on each $p_i$, which jointly rule out $p_i = 2$ and leave $p_i \in \{3,5,7,11,\dots\}$. The three primes are not required to be distinct — $p_1 = p_2 = p_3$ is allowed — and they are unordered in the sense that no inequality relates them. Because a sum of three odd numbers is odd, any $m$ admitting such a representation is necessarily odd, so the assertion implicitly claims that some odd number in the window $[x - 4\cdot 10^{14},\, x-2]$ is a sum of three odd primes; the value $m$ is not claimed to be unique, and nothing is asserted about $x$ itself being such a sum.

The hypotheses are jointly satisfiable — $8{.}7\times 10^{36} \le x \le e^{3100}$ is a non-empty range of naturals, since $e^{3100}$ exceeds $10^{1346}$ — so the statement is not vacuous, and it says nothing at all about $x$ below $87\cdot 10^{35}$ or above $e^{3100}$.