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Statement

Theorem I (p. 103). There are infinitely many distinct positive odd integers that are not of the form $p+2^a+2^b$ with $p$ prime and $a,b>0$. The paper's notation (p. 103) makes all quantities integers, "usually positive integers", and "prime" a positive prime; the theorem itself names positive powers of $2$, so the exponent-zero case is not part of the statement.

Properties of the constructed integers (pp. 105-106). Fix $k=10$ and $G_k=(2^{2^{k}}+1)/(2^{12}\cdot11131+1)$, a proper divisor of the Fermat number $2^{2^{10}}+1$. For each $n>k$ the proof takes the integers $t\le2^{2^n}-1$ that satisfy the simultaneous congruence system (2) of p. 103 together with the system (3) of p. 105,

$$t\equiv0\ \Bigl(\mathrm{mod}\ \frac{2^{2^n}-1}{G_k}\Bigr),\qquad t\equiv-1\pmod{16};$$

by the Chinese remainder theorem there are $v$ or $v+1$ of them, with $v\ge1$. Writing $t=w\prod_{i=0}^{n-1}B_i$ with $B_i=2^{2^i}+1$ for $i\ne k$ and $B_k=(2^{2^k}+1)/G_k$, each such $t$

• is congruent to $-1$ modulo $16$, with $w\equiv1\pmod{16}$ because $\prod B_i\equiv-1\pmod{16}$;
• is divisible by $B_0B_1=15$ (here $B_0=3$, $B_1=5$) and exceeds $15$, as the proof of Lemma II records (p. 105), hence is composite, a consequence the paper does not state;
• is not the sum of a prime and a positive power of $2$, because it satisfies system (2) (p. 106), and so is not a prime plus two equal positive powers of $2$ (footnote 4);
• is not the sum of a prime and two distinct positive powers of $2$, by Lemma II (p. 104) applied with these $B_i$.

The integers obtained for $n+1$ are divisible by $2^{2^n}+1$ and so exceed those obtained for $n$, which are below $2^{2^n}+1$; hence the family is infinite.

Source. R. Crocker, On the sum of a prime and of two powers of two, Pacific J. Math. 36 (1971), no. 1, 103-107; Theorem I on p. 103, Lemmas I and II on p. 104, the proof of Theorem I on pp. 105-106 and the numerical choices on pp. 106-107, read on the page images (PDF pp. 2-6). The copy read is identified on the source card.

Read depth. Claims checked: the statement, Lemma II and the proof of Theorem I were read clause by clause on the page images, and the properties listed above were located in that proof. The proof of Lemma II was read but its arithmetic not re-derived; the numerical verification that the $28$ congruences of (1) cover every residue modulo $720$, and the existence of distinct primes $p_i$ and of the residue $c$ (p. 107), were not re-derived. Nothing here is independently reviewed.

Proof pointer

Pages 103-107. Lemma I (p. 104): for $n\ge3$, $2^{2^n}-1$ is not a prime plus two distinct positive powers of $2$, since for $a>b$ the number $2^{2^n}-1-2^b(2^{a-b}+1)$ is divisible by $2^{2^r}+1$, where $2^r$ is the largest power of $2$ dividing $a-b$, and exceeds it. Lemma II (p. 104) generalizes this to $w\prod_{i<n}B_i\le2^{2^n}-1$ with $n\ge3$, $w\equiv1\pmod{16}$, $B_i\mid2^{2^i}+1$ and $B_i>1$: some $B_r$ with $r<n$ divides $w\prod B_i-2^a-2^b$, and a residue computation modulo $16$, using $w\equiv1$, $\prod B_i\equiv-1$ and $B_r\equiv1,3,5\pmod{16}$, shows the difference is not $B_r$ itself, so it is not prime. The proof of Theorem I (pp. 105-106) combines Lemma II, which comes from the method of the author's earlier note, with the covering-system method of Sierpiński's book, which the paper calls a slight modification of Erdős's: an overlapping congruence system (1) on the exponent, here the $28$ classes listed on p. 106 with least common modulus $720$, is turned by (2) into congruences on $t$ that keep every $t-2^a$ from being prime, with $p_{h+1}=2^{13}-1$ taken on p. 106 and a residue $c$ chosen on p. 107. The paper closes (p. 107) by checking that the primes $p_i$ can be chosen distinct and coprime to $2^{2^n}-1$, and that $16\prod_{i=1}^{h+1}p_i<2^{434}<2^{998}<G_k$.

Dependencies

Lemmas I and II of the same paper, obtained by the method of the author's 1960/61 note in Mathematics Magazine (the paper's reference [1]); the covering-system method of Sierpiński's Elementary Theory of Numbers (its reference [4]), a slight modification of the method of Erdős's 1952 paper in Mat. Lapok (its reference [3]); Dickson's History of the Theory of Numbers for the numerical facts on p. 107 (its reference [2]).

Bears on

• Problem 9: the theorem is the site's negative result behind the problem, which asks whether the odd integers not of the form $p+2^k+2^l$ have positive upper density; the problem's form allows $k,l\ge0$, while the theorem excludes only positive exponents. The constructed integers avoid the zero-exponent forms as well, an observation of this page rather than of the paper: $t=p+2^0+2^0$ would make $t$ a prime plus $2^1$, and $t=p+2^0+2^l$ with $l\ge1$ forces $p=2$ by parity, making $t$ the prime $3$ plus $2^l$; system (2) excludes both. So the set of Problem 9 is infinite; the construction does not decide whether it has positive upper density.
• Problem 10: the constructed integers, through $N=t+1$ and a parity argument, give infinitely many even integers that are not a prime plus at most three powers of $2$, the settled Grechuk variant recorded on that page; the Lean proofs accepted by Conjectures.io re-prove the construction with $k=10$ and the cofactor $2^{12}\cdot11131+1=45592577$ and close the exponent-zero and equal-exponent cases; see the gist card.