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If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1409.2459","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-08T18:41:30Z","cross_cats_sorted":["cs.IT","math.CO","math.IT"],"title_canon_sha256":"37a8a82b2587ee008583a58626a0e2ba2a4ec30aea1ff3bea61f34563683fd1d","abstract_canon_sha256":"c1177bb19fa55ed92297e6cc64e6d8dcdb64f6e42a7263e8fa814384d927102d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:22:29.580125Z","signature_b64":"FpTga+5fu1yq/Gzq8DNAqtueJF53COgfM8RKOQBKOZmT0QhxuC+mwOntbRwk2jOnMnalHcp5VaTmB3VsfKXrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9474d170a79d6a88b9b110cf6c535f4c79c33d33e9fed6a369668fb611dbc22a","last_reissued_at":"2026-05-18T02:22:29.579655Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:22:29.579655Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.CO","math.IT"],"primary_cat":"math.NT","authors_text":"Daniel J. Katz, Philippe Langevin","submitted_at":"2014-09-08T18:41:30Z","abstract_excerpt":"We consider Weil sums of binomials of the form $W_{F,d}(a)=\\sum_{x \\in F} \\psi(x^d-a x)$, where $F$ is a finite field, $\\psi\\colon F\\to {\\mathbb C}$ is the canonical additive character, $\\gcd(d,|F^\\times|)=1$, and $a \\in F^\\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. 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