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Generate ternary quadratic form that represents n and has determinant 1, for determining sum of 3(4) squares for n
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| assert(b,s)=if(!(b), error(Str(s))); | |
| dbg(a,b,c="")=if(getenv("dbg"),print(a,b,c)); | |
| get_tqf(n,vstart)={ | |
| assert(n%4!=0); | |
| assert(n%8!=7); | |
| my(Q,v,b,p,a12); | |
| if(n%4==2, | |
| for(vv=vstart,oo, | |
| if(ispseudoprime((4*vv+1)*n-1), | |
| v=vv; break())); | |
| write("/dev/stderr",v+1); | |
| b=4*v+1; | |
| p=b*n-1; | |
| assert(kronecker(-b,p)==1); | |
| a12=lift(sqrt(Mod(-b,p))); | |
| , | |
| my(c=4-(n%4)); | |
| assert(((c*n-1)/2)%2==1); | |
| for(vv=vstart,oo, | |
| if(ispseudoprime(((8*vv+c)*n-1)/2), | |
| v=vv; break())); | |
| write("/dev/stderr",v+1); | |
| b=8*v+c; | |
| p=(b*n-1)/2; | |
| assert(kronecker(-b,p)==1); | |
| my(r0=lift(sqrt(Mod(-b,p)))); | |
| a12=abs(r0-(1-(r0%2))*p)); | |
| [(b+a12^2)/(b*n-1),a12,1;a12,b*n-1,0;1,0,n]; | |
| } | |
| squares34(n)={ | |
| my(Q,G,V=valuation(n,4),n1=n>>(2*V),res=[],x=[0,0,1]~); | |
| if(n1%8==7,n=sqrtint(n1);if(n%4==0,n-=1);res=[n*2^V];n1-=n^2;); | |
| vstart=if(getenv("vstart"),eval(getenv("vstart")),0); | |
| Q=get_tqf(n1,vstart); | |
| if(type(Q)!="t_MAT",return([])); | |
| assert(qfeval(Q,x)==n1); | |
| assert(matdet(Q)==1); | |
| dbg(" Q=",Q); | |
| G=qflllgram(Q,flag=1); | |
| dbg(" G=",G); | |
| assert(G~*Q*G==matdiagonal([1,1,1])); | |
| res=concat([2^V*i|i<-G^-1*x],res); | |
| abs(res); | |
| } | |
| { | |
| if(getenv("n")!=0, | |
| n=eval(getenv("n")); | |
| sq=squares34(n); | |
| print(getenv("n"),"=norml2(",sq,")"); | |
| assert(norml2(sq)==n); | |
| print("all asserts OK")); | |
| } |
I learned on pari-users list from Bill how to determine all sums of three squares using qfminim():
https://pari.math.u-bordeaux.fr/archives/pari-users-2312/msg00074.html
In S2.b.gp from that posting qflllgram() is not used.
And number of determined vectors is identical to
https://en.wikipedia.org/wiki/Sum_of_squares_function#k_=_3
pi@raspberrypi5:~ $ n=101 gp -q < S2.b.gp
101=norml2([1, 6, 8])
all asserts OK
528979 [711, -153, -7]~ [2, 9, 4]~
528979 [-711, 153, 7]~ [-2, -9, -4]~
10232019 [3127, -673, -31]~ [2, 9, -4]~
10232019 [-3127, 673, 31]~ [-2, -9, 4]~
13150419 [3545, -763, -35]~ [0, -1, 10]~
13150419 [-3545, 763, 35]~ [0, 1, -10]~
43831329 [6472, -1393, -64]~ [-1, -6, 8]~
43831329 [-6472, 1393, 64]~ [1, 6, -8]~
96137219 [9585, -2063, -95]~ [0, -1, -10]~
96137219 [-9585, 2063, 95]~ [0, 1, 10]~
133712449 [11304, -2433, -112]~ [-1, -6, -8]~
133712449 [-11304, 2433, 112]~ [1, 6, 8]~
427236237 [20206, -4349, -200]~ [-1, -8, 6]~
427236237 [-20206, 4349, 200]~ [1, 8, -6]~
594231237 [23830, -5129, -236]~ [-1, -8, -6]~
594231237 [-23830, 5129, 236]~ [1, 8, 6]~
790321533 [27482, -5915, -272]~ [1, 0, 10]~
790321533 [-27482, 5915, 272]~ [-1, 0, -10]~
1073278614 [32026, -6893, -317]~ [2, 4, 9]~
1073278614 [-32026, 6893, 317]~ [-2, -4, -9]~
1175890933 [33522, -7215, -332]~ [1, 0, -10]~
1175890933 [-33522, 7215, 332]~ [-1, 0, 10]~
1292732233 [35148, -7565, -348]~ [-1, -10, 0]~
1292732233 [-35148, 7565, 348]~ [1, 10, 0]~
1468551054 [37462, -8063, -371]~ [2, 4, -9]~
1468551054 [-37462, 8063, 371]~ [-2, -4, 9]~
4063947019 [62319, -13413, -617]~ [4, 9, 2]~
4063947019 [-62319, 13413, 617]~ [-4, -9, -2]~
4223026299 [63527, -13673, -629]~ [4, 9, -2]~
4223026299 [-63527, 13673, 629]~ [-4, -9, 2]~
4468603938 [65348, -14065, -647]~ [0, -10, 1]~
4468603938 [-65348, 14065, 647]~ [0, 10, -1]~
4551590738 [65952, -14195, -653]~ [0, -10, -1]~
4551590738 [-65952, 14195, 653]~ [0, 10, 1]~
4764374329 [67476, -14523, -668]~ [1, -6, 8]~
4764374329 [-67476, 14523, 668]~ [-1, 6, -8]~
5471166489 [72308, -15563, -716]~ [1, -6, -8]~
5471166489 [-72308, 15563, 716]~ [-1, 6, 8]~
5767594747 [74241, -15979, -735]~ [4, 7, 6]~
5767594747 [-74241, 15979, 735]~ [-4, -7, -6]~
6344416747 [77865, -16759, -771]~ [4, 7, -6]~
6344416747 [-77865, 16759, 771]~ [-4, -7, 6]~
6781754154 [80504, -17327, -797]~ [4, 6, 7]~
6781754154 [-80504, 17327, 797]~ [-4, -6, -7]~
6901225957 [81210, -17479, -804]~ [1, -8, 6]~
6901225957 [-81210, 17479, 804]~ [-1, 8, -6]~
7479857494 [84546, -18197, -837]~ [2, -4, 9]~
7479857494 [-84546, 18197, 837]~ [-2, 4, -9]~
7512803914 [84732, -18237, -839]~ [4, 6, -7]~
7512803914 [-84732, 18237, 839]~ [-4, -6, 7]~
7530904237 [84834, -18259, -840]~ [1, -8, -6]~
7530904237 [-84834, 18259, 840]~ [-1, 8, 6]~
8472634894 [89982, -19367, -891]~ [2, -4, -9]~
8472634894 [-89982, 19367, 891]~ [-2, 4, 9]~
9674396433 [96152, -20695, -952]~ [1, -10, 0]~
9674396433 [-96152, 20695, 952]~ [-1, 10, 0]~
11793127002 [106160, -22849, -1051]~ [4, 2, 9]~
11793127002 [-106160, 22849, 1051]~ [-4, -2, -9]~
13031800602 [111596, -24019, -1105]~ [4, 2, -9]~
13031800602 [-111596, 24019, 1105]~ [-4, -2, 9]~
14788772059 [118881, -25587, -1177]~ [2, -9, 4]~
14788772059 [-118881, 25587, 1177]~ [-2, 9, -4]~
15395980059 [121297, -26107, -1201]~ [2, -9, -4]~
15395980059 [-121297, 26107, 1201]~ [-2, 9, 4]~
17736274062 [130190, -28021, -1289]~ [6, 8, 1]~
17736274062 [-130190, 28021, 1289]~ [-6, -8, -1]~
17901226262 [130794, -28151, -1295]~ [6, 8, -1]~
17901226262 [-130794, 28151, 1295]~ [-6, -8, 1]~
18349082122 [132420, -28501, -1311]~ [4, -2, 9]~
18349082122 [-132420, 28501, 1311]~ [-4, 2, -9]~
19311678947 [135849, -29239, -1345]~ [6, 7, 4]~
19311678947 [-135849, 29239, 1345]~ [-6, -7, -4]~
19886508202 [137856, -29671, -1365]~ [4, -2, -9]~
19886508202 [-137856, 29671, 1365]~ [-4, 2, 9]~
20004682467 [138265, -29759, -1369]~ [6, 7, -4]~
20004682467 [-138265, 29759, 1369]~ [-6, -7, 4]~
25023013094 [154638, -33283, -1531]~ [6, 4, 7]~
25023013094 [-154638, 33283, 1531]~ [-6, -4, -7]~
26410041534 [158866, -34193, -1573]~ [6, 4, -7]~
26410041534 [-158866, 34193, 1573]~ [-6, -4, 7]~
26549203674 [159284, -34283, -1577]~ [4, -6, 7]~
26549203674 [-159284, 34283, 1577]~ [-4, 6, -7]~
27977342554 [163512, -35193, -1619]~ [4, -6, -7]~
27977342554 [-163512, 35193, 1619]~ [-4, 6, 7]~
28887709947 [166151, -35761, -1645]~ [4, -7, 6]~
28887709947 [-166151, 35761, 1645]~ [-4, 7, -6]~
30161621067 [169775, -36541, -1681]~ [4, -7, -6]~
30161621067 [-169775, 36541, 1681]~ [-4, 7, 6]~
31287979089 [172916, -37217, -1712]~ [7, 6, 4]~
31287979089 [-172916, 37217, 1712]~ [-7, -6, -4]~
31692784539 [174031, -37457, -1723]~ [6, 1, 8]~
31692784539 [-174031, 37457, 1723]~ [-6, -1, -8]~
32168405089 [175332, -37737, -1736]~ [7, 6, -4]~
32168405089 [-175332, 37737, 1736]~ [-7, -6, 4]~
33477128219 [178863, -38497, -1771]~ [6, 1, -8]~
33477128219 [-178863, 38497, 1771]~ [-6, -1, 8]~
34088561899 [180489, -38847, -1787]~ [4, -9, 2]~
34088561899 [-180489, 38847, 1787]~ [-4, 9, -2]~
34546393659 [181697, -39107, -1799]~ [4, -9, -2]~
34546393659 [-181697, 39107, 1799]~ [-4, 9, 2]~
35985153829 [185442, -39913, -1836]~ [7, 4, 6]~
35985153829 [-185442, 39913, 1836]~ [-7, -4, -6]~
36655393619 [187161, -40283, -1853]~ [6, -1, 8]~
36655393619 [-187161, 40283, 1853]~ [-6, 1, -8]~
37405376989 [189066, -40693, -1872]~ [7, 4, -6]~
37405376989 [-189066, 40693, 1872]~ [-7, -4, 6]~
38572516179 [191993, -41323, -1901]~ [6, -1, -8]~
38572516179 [-191993, 41323, 1901]~ [-6, 1, 8]~
43686366034 [204324, -43977, -2023]~ [8, 6, 1]~
43686366034 [-204324, 43977, 2023]~ [-8, -6, -1]~
43945029474 [204928, -44107, -2029]~ [8, 6, -1]~
43945029474 [-204928, 44107, 2029]~ [-8, -6, 1]~
44906644134 [207158, -44587, -2051]~ [6, -4, 7]~
44906644134 [-207158, 44587, 2051]~ [-6, 4, -7]~
46758398654 [211386, -45497, -2093]~ [6, -4, -7]~
46758398654 [-211386, 45497, 2093]~ [-6, 4, 7]~
54282305547 [227759, -49021, -2255]~ [6, -7, 4]~
54282305547 [-227759, 49021, 2255]~ [-6, 7, -4]~
55440035147 [230175, -49541, -2279]~ [6, -7, -4]~
55440035147 [-230175, 49541, 2279]~ [-6, 7, 4]~
57901872782 [235230, -50629, -2329]~ [6, -8, 1]~
57901872782 [-235230, 50629, 2329]~ [-6, 8, -1]~
58103395299 [235639, -50717, -2333]~ [8, 1, 6]~
58103395299 [-235639, 50717, 2333]~ [-8, -1, -6]~
58199603862 [235834, -50759, -2335]~ [6, -8, -1]~
58199603862 [-235834, 50759, 2335]~ [-6, 8, 1]~
59254645269 [237962, -51217, -2356]~ [7, -4, 6]~
59254645269 [-237962, 51217, 2356]~ [-7, 4, -6]~
59904336339 [239263, -51497, -2369]~ [8, 1, -6]~
59904336339 [-239263, 51497, 2369]~ [-8, -1, 6]~
61073205069 [241586, -51997, -2392]~ [7, -4, -6]~
61073205069 [-241586, 51997, 2392]~ [-7, 4, 6]~
64179725829 [247654, -53303, -2452]~ [9, 4, 2]~
64179725829 [-247654, 53303, 2452]~ [-9, -4, -2]~
64758934579 [248769, -53543, -2463]~ [8, -1, 6]~
64758934579 [-248769, 53543, 2463]~ [-8, 1, -6]~
64807361309 [248862, -53563, -2464]~ [9, 4, -2]~
64807361309 [-248862, 53563, 2464]~ [-9, -4, 2]~
66291800409 [251696, -54173, -2492]~ [7, -6, 4]~
66291800409 [-251696, 54173, 2492]~ [-7, 6, -4]~
66659459779 [252393, -54323, -2499]~ [8, -1, -6]~
66659459779 [-252393, 54323, 2499]~ [-8, 1, 6]~
67570563049 [254112, -54693, -2516]~ [7, -6, -4]~
67570563049 [-254112, 54693, 2516]~ [-7, 6, 4]~
70836156177 [260180, -55999, -2576]~ [9, 2, 4]~
70836156177 [-260180, 55999, 2576]~ [-9, -2, -4]~
72157816577 [262596, -56519, -2600]~ [9, 2, -4]~
72157816577 [-262596, 56519, 2600]~ [-9, -2, 4]~
83868562114 [283104, -60933, -2803]~ [8, -6, 1]~
83868562114 [-283104, 60933, 2803]~ [-8, 6, -1]~
84226809714 [283708, -61063, -2809]~ [8, -6, -1]~
84226809714 [-283708, 61063, 2809]~ [-8, 6, 1]~
85856762297 [286440, -61651, -2836]~ [9, -2, 4]~
85856762297 [-286440, 61651, 2836]~ [-9, 2, -4]~
87311201577 [288856, -62171, -2860]~ [9, -2, -4]~
87311201577 [-288856, 62171, 2860]~ [-9, 2, 4]~
93210511219 [298455, -64237, -2955]~ [10, 1, 0]~
93210511219 [-298455, 64237, 2955]~ [-10, -1, 0]~
94287327509 [300174, -64607, -2972]~ [9, -4, 2]~
94287327509 [-300174, 64607, 2972]~ [-9, 4, -2]~
95047741869 [301382, -64867, -2984]~ [9, -4, -2]~
95047741869 [-301382, 64867, 2984]~ [-9, 4, 2]~
97163554038 [304718, -65585, -3017]~ [10, 0, 1]~
97163554038 [-304718, 65585, 3017]~ [-10, 0, -1]~
97549123438 [305322, -65715, -3023]~ [10, 0, -1]~
97549123438 [-305322, 65715, 3023]~ [-10, 0, 1]~
101592175419 [311585, -67063, -3085]~ [10, -1, 0]~
101592175419 [-311585, 67063, 3085]~ [-10, 1, 0]~
#S2=168
12*h(-4*n)=168
101592175419 [311585, -67063, -3085]~ [10, -1, 0]~
pi@raspberrypi5:~ $
With this little diff using "qflllgram()" ...
pi@raspberrypi5:~ $ diff S2.b.gp S2.c.gp
3c3
< Q=get_tqf(n);
---
> Q=qflllgram(get_tqf(n))^-1;
pi@raspberrypi5:~ $
... the vector
pi@raspberrypi5:~ $ n=101 gp -q < S2.c.gp
101=norml2([1, 6, 8])
all asserts OK
1 [0, 0, 1]~ [-1, -6, -8]~
1 [0, 0, -1]~ [1, 6, 8]~
651 [25, -5, -1]~ [6, 1, 8]~
651 [-25, 5, 1]~ [-6, -1, -8]~
6886 [81, -18, -1]~ [-8, 6, -1]~
6886 [-81, 18, 1]~ [8, -6, 1]~
23826 [151, -32, -1]~ [6, -8, -1]~
23826 [-151, 32, 1]~ [-6, 8, 1]~
23891 [151, -33, -1]~ [-8, 1, -6]~
23891 [-151, 33, 1]~ [8, -1, 6]~
40181 [196, -42, -1]~ [1, -8, -6]~
40181 [-196, 42, 1]~ [-1, 8, 6]~
43561 [204, -44, -3]~ [-1, 6, 8]~
43561 [-204, 44, 3]~ [1, -6, -8]~
43970 [205, -44, -3]~ [2, 4, 9]~
43970 [-205, 44, 3]~ [-2, -4, -9]~
48350 [215, -46, -3]~ [4, 2, 9]~
48350 [-215, 46, 3]~ [-4, -2, -9]~
49307 [217, -47, -3]~ [-4, 7, 6]~
49307 [-217, 47, 3]~ [4, -7, -6]~
57835 [235, -51, -3]~ [-6, 7, 4]~
57835 [-235, 51, 3]~ [6, -7, -4]~
59731 [239, -51, -3]~ [6, -1, 8]~
59731 [-239, 51, 3]~ [-6, 1, -8]~
74662 [267, -58, -3]~ [-8, 6, 1]~
74662 [-267, 58, 3]~ [8, -6, -1]~
77357 [272, -58, -3]~ [7, -4, 6]~
77357 [-272, 58, 3]~ [-7, 4, -6]~
94105 [300, -64, -3]~ [7, -6, 4]~
94105 [-300, 64, 3]~ [-7, 6, -4]~
96781 [304, -66, -3]~ [-9, 4, -2]~
96781 [-304, 66, 3]~ [9, -4, 2]~
115417 [332, -72, -3]~ [-9, 2, -4]~
115417 [-332, 72, 3]~ [9, -2, 4]~
118762 [337, -72, -3]~ [6, -8, 1]~
118762 [-337, 72, 3]~ [-6, 8, -1]~
139475 [365, -79, -3]~ [-8, -1, -6]~
139475 [-365, 79, 3]~ [8, 1, 6]~
142411 [369, -79, -3]~ [4, -9, -2]~
142411 [-369, 79, 3]~ [-4, 9, 2]~
156667 [387, -83, -3]~ [2, -9, -4]~
156667 [-387, 83, 3]~ [-2, 9, 4]~
158386 [389, -84, -3]~ [-6, -4, -7]~
158386 [-389, 84, 3]~ [6, 4, 7]~
166606 [399, -86, -3]~ [-4, -6, -7]~
166606 [-399, 86, 3]~ [4, 6, 7]~
167405 [400, -86, -3]~ [-1, -8, -6]~
167405 [-400, 86, 3]~ [1, 8, 6]~
182014 [417, -90, -5]~ [-4, 6, 7]~
182014 [-417, 90, 5]~ [4, -6, -7]~
206377 [444, -96, -5]~ [-7, 6, 4]~
206377 [-444, 96, 5]~ [7, -6, -4]~
228114 [467, -100, -5]~ [6, -4, 7]~
228114 [-467, 100, 5]~ [-6, 4, -7]~
270987 [509, -109, -5]~ [6, -7, 4]~
270987 [-509, 109, 5]~ [-6, 7, -4]~
356957 [584, -126, -5]~ [-7, -4, -6]~
356957 [-584, 126, 5]~ [7, 4, 6]~
375467 [599, -129, -5]~ [-4, -7, -6]~
375467 [-599, 129, 5]~ [4, 7, 6]~
393242 [613, -132, -7]~ [-2, 4, 9]~
393242 [-613, 132, 7]~ [2, -4, -9]~
432542 [643, -138, -7]~ [4, -2, 9]~
432542 [-643, 138, 7]~ [-4, 2, -9]~
478341 [676, -146, -7]~ [-9, 4, 2]~
478341 [-676, 146, 7]~ [9, -4, -2]~
574411 [741, -159, -7]~ [4, -9, 2]~
574411 [-741, 159, 7]~ [-4, 9, -2]~
604545 [760, -164, -7]~ [-9, -2, -4]~
604545 [-760, 164, 7]~ [9, 2, 4]~
661315 [795, -171, -7]~ [-2, -9, -4]~
661315 [-795, 171, 7]~ [2, 9, 4]~
708739 [823, -177, -9]~ [0, 1, 10]~
708739 [-823, 177, 9]~ [0, -1, -10]~
717349 [828, -178, -9]~ [1, 0, 10]~
717349 [-828, 178, 9]~ [-1, 0, -10]~
729706 [835, -180, -9]~ [-6, 4, 7]~
729706 [-835, 180, 9]~ [6, -4, -7]~
745541 [844, -182, -9]~ [-7, 4, 6]~
745541 [-844, 182, 9]~ [7, -4, -6]~
819406 [885, -190, -9]~ [4, -6, 7]~
819406 [-885, 190, 9]~ [-4, 6, -7]~
845531 [899, -193, -9]~ [4, -7, 6]~
845531 [-899, 193, 9]~ [-4, 7, -6]~
872459 [913, -197, -9]~ [-10, 1, 0]~
872459 [-913, 197, 9]~ [10, -1, 0]~
899410 [927, -200, -9]~ [-10, 0, -1]~
899410 [-927, 200, 9]~ [10, 0, 1]~
980369 [968, -208, -9]~ [1, -10, 0]~
980369 [-968, 208, 9]~ [-1, 10, 0]~
998710 [977, -210, -9]~ [0, -10, -1]~
998710 [-977, 210, 9]~ [0, 10, 1]~
1013281 [984, -212, -9]~ [-7, -6, -4]~
1013281 [-984, 212, 9]~ [7, 6, 4]~
1023571 [989, -213, -9]~ [-6, -7, -4]~
1023571 [-989, 213, 9]~ [6, 7, 4]~
1112366 [1031, -222, -11]~ [-4, 2, 9]~
1112366 [-1031, 222, 11]~ [4, -2, -9]~
1114429 [1032, -222, -11]~ [-1, 0, 10]~
1114429 [-1032, 222, 11]~ [1, 0, -10]~
1125219 [1037, -223, -11]~ [0, -1, 10]~
1125219 [-1037, 223, 11]~ [0, 1, -10]~
1177826 [1061, -228, -11]~ [2, -4, 9]~
1177826 [-1061, 228, 11]~ [-2, 4, -9]~
1211721 [1076, -232, -11]~ [-9, 2, 4]~
1211721 [-1076, 232, 11]~ [9, -2, -4]~
1296490 [1113, -240, -11]~ [-10, 0, 1]~
1296490 [-1113, 240, 11]~ [10, 0, -1]~
1329299 [1127, -243, -11]~ [-10, -1, 0]~
1329299 [-1127, 243, 11]~ [10, 1, 0]~
1338331 [1131, -243, -11]~ [2, -9, 4]~
1338331 [-1131, 243, 11]~ [-2, 9, -4]~
1408221 [1160, -250, -11]~ [-9, -4, -2]~
1408221 [-1160, 250, 11]~ [9, 4, 2]~
1415190 [1163, -250, -11]~ [0, -10, 1]~
1415190 [-1163, 250, 11]~ [0, 10, -1]~
1437209 [1172, -252, -11]~ [-1, -10, 0]~
1437209 [-1172, 252, 11]~ [1, 10, 0]~
1469371 [1185, -255, -11]~ [-4, -9, -2]~
1469371 [-1185, 255, 11]~ [4, 9, 2]~
1632531 [1249, -269, -13]~ [-6, 1, 8]~
1632531 [-1249, 269, 13]~ [6, -1, -8]~
1679987 [1267, -273, -13]~ [-8, 1, 6]~
1679987 [-1267, 273, 13]~ [8, -1, -6]~
1725001 [1284, -276, -13]~ [1, -6, 8]~
1725001 [-1284, 276, 13]~ [-1, 6, -8]~
1801037 [1312, -282, -13]~ [1, -8, 6]~
1801037 [-1312, 282, 13]~ [-1, 8, -6]~
1949830 [1365, -294, -13]~ [-8, -6, -1]~
1949830 [-1365, 294, 13]~ [8, 6, 1]~
1978410 [1375, -296, -13]~ [-6, -8, -1]~
1978410 [-1375, 296, 13]~ [6, 8, 1]~
2227502 [1459, -314, -15]~ [-4, -2, 9]~
2227502 [-1459, 314, 15]~ [4, 2, -9]~
2239819 [1463, -315, -15]~ [-6, -1, 8]~
2239819 [-1463, 315, 15]~ [6, 1, -8]~
2258042 [1469, -316, -15]~ [-2, -4, 9]~
2258042 [-1469, 316, 15]~ [2, 4, -9]~
2295347 [1481, -319, -15]~ [-8, -1, 6]~
2295347 [-1481, 319, 15]~ [8, 1, -6]~
2316769 [1488, -320, -15]~ [-1, -6, 8]~
2316769 [-1488, 320, 15]~ [1, 6, -8]~
2367217 [1504, -324, -15]~ [-9, -2, 4]~
2367217 [-1504, 324, 15]~ [9, 2, -4]~
2404757 [1516, -326, -15]~ [-1, -8, 6]~
2404757 [-1516, 326, 15]~ [1, 8, -6]~
2456149 [1532, -330, -15]~ [-9, -4, 2]~
2456149 [-1532, 330, 15]~ [9, 4, -2]~
2478307 [1539, -331, -15]~ [-2, -9, 4]~
2478307 [-1539, 331, 15]~ [2, 9, -4]~
2517382 [1551, -334, -15]~ [-8, -6, 1]~
2517382 [-1551, 334, 15]~ [8, 6, -1]~
2536699 [1557, -335, -15]~ [-4, -9, 2]~
2536699 [-1557, 335, 15]~ [4, 9, -2]~
2549842 [1561, -336, -15]~ [-6, -8, 1]~
2549842 [-1561, 336, 15]~ [6, 8, -1]~
2992266 [1691, -364, -17]~ [-6, -4, 7]~
2992266 [-1691, 364, 17]~ [6, 4, -7]~
3024245 [1700, -366, -17]~ [-7, -4, 6]~
3024245 [-1700, 366, 17]~ [7, 4, -6]~
3027646 [1701, -366, -17]~ [-4, -6, 7]~
3027646 [-1701, 366, 17]~ [4, 6, -7]~
3077675 [1715, -369, -17]~ [-4, -7, 6]~
3077675 [-1715, 369, 17]~ [4, 7, -6]~
3124657 [1728, -372, -17]~ [-7, -6, 4]~
3124657 [-1728, 372, 17]~ [7, 6, -4]~
3142707 [1733, -373, -17]~ [-6, -7, 4]~
3142707 [-1733, 373, 17]~ [6, 7, -4]~
#S2=168
12*h(-4*n)=168
708739 [823, -177, -9]~ [0, 1, 10]~
pi@raspberrypi5:~ $
Before the get_tqf() loops always started at 0, now they start at vstart environment variable.
This allows to get more than 1 solution for a given n.
Next vstart start value for (different) output is printed to /dev/stderr:
pi@raspberrypi5:~ $ n=101 gp -q < tqf.gp
1
101=norml2([1, 6, 8])
all asserts OK
pi@raspberrypi5:~ $ vstart=1 n=101 gp -q < tqf.gp
13
101=norml2([6, 4, 7])
all asserts OK
pi@raspberrypi5:~ $ vstart=13 n=101 gp -q < tqf.gp
15
101=norml2([9, 4, 2])
all asserts OK
pi@raspberrypi5:~ $ vstart=15 n=101 gp -q < tqf.gp
16
101=norml2([0, 10, 1])
all asserts OK
pi@raspberrypi5:~ $
Redirecting 3 to standard output allows to always use the same command for iteration over possible sum of squares:
pi@raspberrypi5:~ $ exec 3>&1
pi@raspberrypi5:~ $ vstart=
pi@raspberrypi5:~ $ vstart=$(vstart=$vstart n=101 gp -q < tqf.gp 2>&1 1>&3)
101=norml2([1, 6, 8])
all asserts OK
pi@raspberrypi5:~ $ vstart=$(vstart=$vstart n=101 gp -q < tqf.gp 2>&1 1>&3)
101=norml2([6, 4, 7])
all asserts OK
pi@raspberrypi5:~ $ vstart=$(vstart=$vstart n=101 gp -q < tqf.gp 2>&1 1>&3)
101=norml2([9, 4, 2])
all asserts OK
pi@raspberrypi5:~ $ vstart=$(vstart=$vstart n=101 gp -q < tqf.gp 2>&1 1>&3)
101=norml2([0, 10, 1])
all asserts OK
pi@raspberrypi5:~ $
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Last update was cosmetic, to make printed output correct.
Before:
Now: