History of Fekete polynomials $f_p$

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when	who	what
2026-09-18 01:20	zeta3 table-build@2.0+395f185d	shorten the Fekete definition	current reviewed
2026-09-18 01:19	zeta3 table-build@2.0+395f185d	attach Fekete polynomial generator
2026-09-18 01:19	zeta3 table-build@2.0+395f185d	fill Fekete polynomial entries
2026-09-18 01:15	zeta3 table-build@2.0+395f185d	claim Fekete polynomials

What changed between 2026-09-18 01:19 and 2026-09-18 01:20

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 Definition: For an odd prime $p$, the Fekete polynomial is $f_p(x)=\sum_{a=0}^{p-1}\left(\frac{a}{p}\right)x^a$,   where $\left(\frac{a}{p}\right)$ is the Legendre symbol CITE{GS} CITE{Legendre}.-  The variable is $x$, the constant term is $0$, and the coefficient of $x^a$ is $1$-  for a nonzero quadratic residue modulo $p$ and $-1$ for a quadratic nonresidue modulo-  $p$. Parameters:   p:

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     $0$. The Legendre sequence of length $p$ instead puts $u_0=1$; that is a different     object.+  comment-coefficients: This table uses $x$ as the variable. The constant term is+    $0$, and the coefficient of $x^a$ is $1$ for a nonzero quadratic residue modulo+    $p$ and $-1$ for a quadratic nonresidue modulo $p$. Formulas:   formula-sum-at-one: $f_p(1)=0$, since the nonzero residues and nonresidues modulo

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   number-header: $f_p(x)$ Numbers:-- params:-    p: '3'-  number: -x^2 + x-- params:-    p: '5'-  number: x^4 - x^3 - x^2 + x-- params:-    p: '7'-  number: -x^6 - x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '11'-  number: -x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + x^4 + x^3 - x^2 + x-- params:-    p: '13'-  number: x^12 - x^11 + x^10 + x^9 - x^8 - x^7 - x^6 - x^5 + x^4 + x^3 - x^2 + x-- params:-    p: '17'-  number: x^16 + x^15 - x^14 + x^13 - x^12 - x^11 - x^10 + x^9 + x^8 - x^7 - x^6 --    x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '19'-  number: -x^18 + x^17 + x^16 - x^15 - x^14 - x^13 - x^12 + x^11 - x^10 + x^9 - x^8+  '3': -x^2 + x+  '5': x^4 - x^3 - x^2 + x+  '7': -x^6 - x^5 + x^4 - x^3 + x^2 + x+  '11': -x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + x^4 + x^3 - x^2 + x+  '13': x^12 - x^11 + x^10 + x^9 - x^8 - x^7 - x^6 - x^5 + x^4 + x^3 - x^2 + x+  '17': x^16 + x^15 - x^14 + x^13 - x^12 - x^11 - x^10 + x^9 + x^8 - x^7 - x^6 - x^5+    + x^4 - x^3 + x^2 + x+  '19': -x^18 + x^17 + x^16 - x^15 - x^14 - x^13 - x^12 + x^11 - x^10 + x^9 - x^8     + x^7 + x^6 + x^5 + x^4 - x^3 - x^2 + x-- params:-    p: '23'-  number: -x^22 - x^21 - x^20 - x^19 + x^18 - x^17 + x^16 - x^15 - x^14 + x^13 + x^12+  '23': -x^22 - x^21 - x^20 - x^19 + x^18 - x^17 + x^16 - x^15 - x^14 + x^13 + x^12     - x^11 - x^10 + x^9 + x^8 - x^7 + x^6 - x^5 + x^4 + x^3 + x^2 + x-- params:-    p: '29'-  number: x^28 - x^27 - x^26 + x^25 + x^24 + x^23 + x^22 - x^21 + x^20 - x^19 - x^18+  '29': x^28 - x^27 - x^26 + x^25 + x^24 + x^23 + x^22 - x^21 + x^20 - x^19 - x^18     - x^17 + x^16 - x^15 - x^14 + x^13 - x^12 - x^11 - x^10 + x^9 - x^8 + x^7 + x^6     + x^5 + x^4 - x^3 - x^2 + x-- params:-    p: '31'-  number: -x^30 - x^29 + x^28 - x^27 - x^26 + x^25 - x^24 - x^23 - x^22 - x^21 + x^20+  '31': -x^30 - x^29 + x^28 - x^27 - x^26 + x^25 - x^24 - x^23 - x^22 - x^21 + x^20     + x^19 + x^18 - x^17 + x^16 - x^15 + x^14 - x^13 - x^12 - x^11 + x^10 + x^9 +     x^8 + x^7 - x^6 + x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '37'-  number: x^36 - x^35 + x^34 + x^33 - x^32 - x^31 + x^30 - x^29 + x^28 + x^27 + x^26+  '37': x^36 - x^35 + x^34 + x^33 - x^32 - x^31 + x^30 - x^29 + x^28 + x^27 + x^26     + x^25 - x^24 - x^23 - x^22 + x^21 - x^20 - x^19 - x^18 - x^17 + x^16 - x^15 -     x^14 - x^13 + x^12 + x^11 + x^10 + x^9 - x^8 + x^7 - x^6 - x^5 + x^4 + x^3 - x^2     + x-- params:-    p: '41'-  number: x^40 + x^39 - x^38 + x^37 + x^36 - x^35 - x^34 + x^33 + x^32 + x^31 - x^30+  '41': x^40 + x^39 - x^38 + x^37 + x^36 - x^35 - x^34 + x^33 + x^32 + x^31 - x^30     - x^29 - x^28 - x^27 - x^26 + x^25 - x^24 + x^23 - x^22 + x^21 + x^20 - x^19 +     x^18 - x^17 + x^16 - x^15 - x^14 - x^13 - x^12 - x^11 + x^10 + x^9 + x^8 - x^7     - x^6 + x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '43'-  number: -x^42 + x^41 + x^40 - x^39 + x^38 - x^37 + x^36 + x^35 - x^34 - x^33 - x^32+  '43': -x^42 + x^41 + x^40 - x^39 + x^38 - x^37 + x^36 + x^35 - x^34 - x^33 - x^32     + x^31 - x^30 - x^29 - x^28 - x^27 - x^26 + x^25 + x^24 + x^23 - x^22 + x^21 -     x^20 - x^19 - x^18 + x^17 + x^16 + x^15 + x^14 + x^13 - x^12 + x^11 + x^10 + x^9     - x^8 - x^7 + x^6 - x^5 + x^4 - x^3 - x^2 + x-- params:-    p: '47'-  number: -x^46 - x^45 - x^44 - x^43 + x^42 - x^41 - x^40 - x^39 - x^38 + x^37 + x^36+  '47': -x^46 - x^45 - x^44 - x^43 + x^42 - x^41 - x^40 - x^39 - x^38 + x^37 + x^36     - x^35 + x^34 - x^33 + x^32 - x^31 - x^30 - x^29 + x^28 + x^27 - x^26 + x^25 +     x^24 - x^23 - x^22 + x^21 - x^20 - x^19 + x^18 + x^17 + x^16 - x^15 + x^14 - x^13     + x^12 - x^11 - x^10 + x^9 + x^8 + x^7 + x^6 - x^5 + x^4 + x^3 + x^2 + x-- params:-    p: '53'-  number: x^52 - x^51 - x^50 + x^49 - x^48 + x^47 + x^46 - x^45 + x^44 + x^43 + x^42+  '53': x^52 - x^51 - x^50 + x^49 - x^48 + x^47 + x^46 - x^45 + x^44 + x^43 + x^42     - x^41 + x^40 - x^39 + x^38 + x^37 + x^36 - x^35 - x^34 - x^33 - x^32 - x^31 -     x^30 + x^29 + x^28 - x^27 - x^26 + x^25 + x^24 - x^23 - x^22 - x^21 - x^20 - x^19     - x^18 + x^17 + x^16 + x^15 - x^14 + x^13 - x^12 + x^11 + x^10 + x^9 - x^8 + x^7     + x^6 - x^5 + x^4 - x^3 - x^2 + x-- params:-    p: '59'-  number: -x^58 + x^57 - x^56 - x^55 - x^54 + x^53 - x^52 + x^51 - x^50 + x^49 + x^48+  '59': -x^58 + x^57 - x^56 - x^55 - x^54 + x^53 - x^52 + x^51 - x^50 + x^49 + x^48     - x^47 + x^46 + x^45 - x^44 - x^43 - x^42 + x^41 - x^40 - x^39 - x^38 - x^37 +     x^36 + x^35 - x^34 - x^33 - x^32 - x^31 - x^30 + x^29 + x^28 + x^27 + x^26 + x^25     - x^24 - x^23 + x^22 + x^21 + x^20 + x^19 - x^18 + x^17 + x^16 + x^15 - x^14 -     x^13 + x^12 - x^11 - x^10 + x^9 - x^8 + x^7 - x^6 + x^5 + x^4 + x^3 - x^2 + x-- params:-    p: '61'-  number: x^60 - x^59 + x^58 + x^57 + x^56 - x^55 - x^54 - x^53 + x^52 - x^51 - x^50+  '61': x^60 - x^59 + x^58 + x^57 + x^56 - x^55 - x^54 - x^53 + x^52 - x^51 - x^50     + x^49 + x^48 + x^47 + x^46 + x^45 - x^44 - x^43 + x^42 + x^41 - x^40 + x^39 -     x^38 - x^37 + x^36 - x^35 + x^34 - x^33 - x^32 - x^31 - x^30 - x^29 - x^28 + x^27

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     x^15 + x^14 + x^13 + x^12 - x^11 - x^10 + x^9 - x^8 - x^7 - x^6 + x^5 + x^4 +     x^3 - x^2 + x-- params:-    p: '67'-  number: -x^66 + x^65 + x^64 - x^63 + x^62 - x^61 + x^60 + x^59 - x^58 - x^57 + x^56+  '67': -x^66 + x^65 + x^64 - x^63 + x^62 - x^61 + x^60 + x^59 - x^58 - x^57 + x^56     + x^55 + x^54 - x^53 - x^52 - x^51 - x^50 + x^49 - x^48 + x^47 - x^46 - x^45 -     x^44 - x^43 - x^42 - x^41 + x^40 + x^39 - x^38 + x^37 + x^36 + x^35 - x^34 + x^33

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     x^21 - x^20 + x^19 - x^18 + x^17 + x^16 + x^15 + x^14 - x^13 - x^12 - x^11 + x^10     + x^9 - x^8 - x^7 + x^6 - x^5 + x^4 - x^3 - x^2 + x-- params:-    p: '71'-  number: -x^70 - x^69 - x^68 - x^67 - x^66 - x^65 + x^64 - x^63 - x^62 - x^61 + x^60+  '71': -x^70 - x^69 - x^68 - x^67 - x^66 - x^65 + x^64 - x^63 - x^62 - x^61 + x^60     - x^59 + x^58 + x^57 - x^56 - x^55 + x^54 - x^53 - x^52 - x^51 + x^50 + x^49 +     x^48 - x^47 - x^46 + x^45 - x^44 + x^43 - x^42 - x^41 + x^40 - x^39 + x^38 + x^37

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     - x^13 + x^12 - x^11 + x^10 + x^9 + x^8 - x^7 + x^6 + x^5 + x^4 + x^3 + x^2 +     x-- params:-    p: '73'-  number: x^72 + x^71 + x^70 + x^69 - x^68 + x^67 - x^66 + x^65 + x^64 - x^63 - x^62+  '73': x^72 + x^71 + x^70 + x^69 - x^68 + x^67 - x^66 + x^65 + x^64 - x^63 - x^62     + x^61 - x^60 - x^59 - x^58 + x^57 - x^56 + x^55 + x^54 - x^53 - x^52 - x^51 +     x^50 + x^49 + x^48 - x^47 + x^46 - x^45 - x^44 - x^43 - x^42 + x^41 - x^40 - x^39

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     - x^15 - x^14 - x^13 + x^12 - x^11 - x^10 + x^9 + x^8 - x^7 + x^6 - x^5 + x^4     + x^3 + x^2 + x-- params:-    p: '79'-  number: -x^78 - x^77 + x^76 - x^75 - x^74 + x^73 + x^72 - x^71 - x^70 - x^69 - x^68+  '79': -x^78 - x^77 + x^76 - x^75 - x^74 + x^73 + x^72 - x^71 - x^70 - x^69 - x^68     + x^67 - x^66 + x^65 + x^64 - x^63 + x^62 - x^61 - x^60 - x^59 - x^58 - x^57 -     x^56 + x^55 - x^54 - x^53 + x^52 + x^51 + x^50 + x^49 - x^48 - x^47 + x^46 + x^45

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     + x^21 + x^20 + x^19 + x^18 - x^17 + x^16 - x^15 - x^14 + x^13 - x^12 + x^11 +     x^10 + x^9 + x^8 - x^7 - x^6 + x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '83'-  number: -x^82 + x^81 - x^80 - x^79 + x^78 + x^77 - x^76 + x^75 - x^74 - x^73 - x^72+  '83': -x^82 + x^81 - x^80 - x^79 + x^78 + x^77 - x^76 + x^75 - x^74 - x^73 - x^72     - x^71 + x^70 + x^69 + x^68 - x^67 - x^66 + x^65 + x^64 + x^63 - x^62 + x^61 -     x^60 + x^59 - x^58 - x^57 - x^56 - x^55 - x^54 - x^53 - x^52 + x^51 - x^50 + x^49

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     x^14 - x^13 + x^12 + x^11 + x^10 + x^9 - x^8 + x^7 - x^6 - x^5 + x^4 + x^3 - x^2     + x-- params:-    p: '89'-  number: x^88 + x^87 - x^86 + x^85 + x^84 - x^83 - x^82 + x^81 + x^80 + x^79 + x^78+  '89': x^88 + x^87 - x^86 + x^85 + x^84 - x^83 - x^82 + x^81 + x^80 + x^79 + x^78     - x^77 - x^76 - x^75 - x^74 + x^73 + x^72 + x^71 - x^70 + x^69 + x^68 + x^67 -     x^66 - x^65 + x^64 - x^63 - x^62 - x^61 - x^60 - x^59 - x^58 + x^57 - x^56 + x^55

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     x^20 - x^19 + x^18 + x^17 + x^16 - x^15 - x^14 - x^13 - x^12 + x^11 + x^10 + x^9     + x^8 - x^7 - x^6 + x^5 + x^4 - x^3 + x^2 + x-- params:-    p: '97'-  number: x^96 + x^95 + x^94 + x^93 - x^92 + x^91 - x^90 + x^89 + x^88 - x^87 + x^86+  '97': x^96 + x^95 + x^94 + x^93 - x^92 + x^91 - x^90 + x^89 + x^88 - x^87 + x^86     + x^85 - x^84 - x^83 - x^82 + x^81 - x^80 + x^79 - x^78 - x^77 - x^76 + x^75 -     x^74 + x^73 + x^72 - x^71 + x^70 - x^69 - x^68 - x^67 + x^66 + x^65 + x^64 - x^63

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     + x^16 - x^15 - x^14 - x^13 + x^12 + x^11 - x^10 + x^9 + x^8 - x^7 + x^6 - x^5     + x^4 + x^3 + x^2 + x-- params:-    p: '101'-  number: x^100 - x^99 - x^98 + x^97 + x^96 + x^95 - x^94 - x^93 + x^92 - x^91 - x^90+  '101': x^100 - x^99 - x^98 + x^97 + x^96 + x^95 - x^94 - x^93 + x^92 - x^91 - x^90     - x^89 + x^88 + x^87 - x^86 + x^85 + x^84 - x^83 + x^82 + x^81 + x^80 + x^79 +     x^78 + x^77 + x^76 - x^75 - x^74 - x^73 - x^72 + x^71 + x^70 - x^69 + x^68 - x^67