Let \(\alpha\) be the residue class of \(x\) in \[ \mathbf F_2[x]/(x^{61}+x^{45}+x^{32}+x^2+1). \] For \(0<a<b\), divisibility by \(f\) is equivalent to \[ \alpha^b+\alpha^a+1=0, \] or \(\alpha^a=\alpha^b+1\).

The certificate stores every tagged residue \((\alpha^a,a)\) for \[ 1\leq a<2^{28}. \] It then tests \(\alpha^b+1\) in increasing order for every \(1\leq b\leq2^{28}\), scans the complete bucket containing that residue, and accepts only a matching tag with \(a<b\). It inspected 536,886,339 packed records and found no match. Thus \[ f(x)\nmid x^b+x^a+1 \] for every \(0<a<b\leq268435456\).

This proves one eighth of the requested degree interval. The 1,879,048,192 possible values of \(b\) satisfying \(2^{28}<b\leq2^{31}\) remain outside the certified search.

Write \[ f=x^{61}+x^{45}+x^{32}+x^2+1. \] The exact replay computes \(x^{2^{61}}\bmod f=x\) and \[ \gcd(x^2+x,f)=1. \] Since 61 is prime, these are the two conditions in the finite-field irreducibility criterion for a degree-61 polynomial over \(\mathbf F_2\). Hence \(f\) is irreducible.

The same replay applies the Lucas-Lehmer test to \[ M=2^{61}-1=2305843009213693951 \] and obtains final residue zero, certifying that \(M\) is prime. A root \(\alpha\) of \(f\) is nonzero, and \(f(1)=1\), so \(\alpha\neq1\). Its multiplicative order divides the prime \(M\) and exceeds one. Therefore \(\alpha\) has order \(M\), and \(f\) is primitive.

Didier and Laigle-Chapuy formulate low-weight multiple searches through discrete logarithms and time-memory tradeoffs. Peterlongo, Sala, and Tinnirello state the exact identity used here: if \(\alpha\) is a root of a primitive polynomial, then \[ Z_\alpha(i)=j\quad\Longleftrightarrow\quad 1+\alpha^i=\alpha^j \quad\Longleftrightarrow\quad p\mid1+x^i+x^j. \] Their heuristic critical-degree estimate for weight three is on the scale of \(2^{n/2}\). At \(n=61\), the requested upper bound lies in that birthday range. The estimate is statistical and supplies no finite exclusion.

Maitra, Gupta, and Venkateswarlu count sparse multiples of primitive polynomials and study their degree distribution. Giesbrecht, Roche, and Tilak place sparse multiples over finite fields in a broader algorithmic setting. Focused searches using the exact polynomial, its exponent set \(\{61,45,32,2,0\}\), degree-61 trinomial multiples, and Zech-logarithm tables found no source reporting the least pair for this polynomial.

A full run of the same bucket layout at \(2^{31}\) would require about 25.8 GB before allocator and operating-system overhead. A lower-memory completion could partition the high residue bits, make several sequential passes over the LFSR orbit, and preserve the same exponent tags and per-partition hashes. This entry makes no claim for \(268435456<b\leq2147483648\).

A field element is a 61-bit polynomial residue. Multiplication by \(x\) shifts the word once. When bit 60 leaves the word, reduction by \(f\) xors bits 45, 32, 2, and 0.

The table uses the high 28 residue bits as a bucket number. Each 64-bit record packs the lower 33 residue bits with the 28-bit exponent \(a\). A 32-bit directory identifies the complete contiguous run for each bucket. This retains enough information to test both residue equality and \(a<b\). The table occupies about 3.2 GB.

SHA-256 covers the full little-endian directory and record arrays. Their digests are `dd87c5d9889c5dac153dded50f97c3902c6800ec902b55cff2ca0888d615b3e4` and `46f09e159b66fcf43a035c720fcf15e4861a76f71119ec6033328931b0089fe0`. The four-line output has SHA-256 digest `9f2c423fc725b1f7e83e91635618f2fde48d13b398bb5e3fb6704b46802d770a`.

The program performs carryless polynomial arithmetic with Python integers. It records all 61 Frobenius residues and all 59 Lucas-Lehmer residues in canonical compact JSON hashes. The canonical report has SHA-256 digest `05a362eabd4cd990bfecb9aed0438b10d27a4391c450941798abbe47daa36aa6`.