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| \\ a b c d e | |
| \\ l m n f o | |
| \\ k p g u v | |
| \\ j h x r y | |
| \\ i q w t s | |
| \\ps=[3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 71, 73, 79, 83, 101, 103, 107, 109, 113]; | |
| ps=[3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113]; | |
| used=Vec(0,25); | |
| { | |
| for(a=1,25,if(!used[a], | |
| used[a]=1; | |
| for(b=1,25,if(!used[b], | |
| used[b]=1; | |
| for(c=1,25,if(!used[c], | |
| used[c]=1; | |
| for(d=1,25,if(!used[d], | |
| used[d]=1; | |
| for(e=1,25,if(!used[e], | |
| used[e]=1; | |
| if(isprime(ps[a]+ps[b]+ps[c]+ps[d]+ps[e]), | |
| for(f=1,25,if(!used[f], | |
| used[f]=1; | |
| for(g=1,25,if(!used[g], | |
| used[g]=1; | |
| for(h=1,25,if(!used[h], | |
| used[h]=1; | |
| for(i=1,25,if(!used[i], | |
| used[i]=1; | |
| if(isprime(ps[e]+ps[f]+ps[g]+ps[h]+ps[i]), | |
| for(j=1,25,if(!used[j], | |
| used[j]=1; | |
| for(k=1,25,if(!used[k], | |
| used[k]=1; | |
| for(l=1,25,if(!used[l], | |
| used[l]=1; | |
| if(isprime(ps[i]+ps[j]+ps[k]+ps[l]+ps[a]), | |
| for(m=1,25,if(!used[m], | |
| used[m]=1; | |
| for(n=1,25,if(!used[n], | |
| used[n]=1; | |
| for(o=1,25,if(!used[o], | |
| used[o]=1; | |
| if(isprime(ps[l]+ps[m]+ps[n]+ps[f]+ps[o]), | |
| for(p=1,25,if(!used[p], | |
| used[p]=1; | |
| for(q=1,25,if(!used[q], | |
| used[q]=1; | |
| if(isprime(ps[b]+ps[m]+ps[p]+ps[h]+ps[q]), | |
| for(r=1,25,if(!used[r], | |
| used[r]=1; | |
| for(s=1,25,if(!used[s], | |
| used[s]=1; | |
| if(isprime(ps[a]+ps[m]+ps[g]+ps[r]+ps[s]), | |
| for(t=1,25,if(!used[t], | |
| used[t]=1; | |
| for(u=1,25,if(!used[u], | |
| used[u]=1; | |
| if(isprime(ps[t]+ps[r]+ps[u]+ps[f]+ps[d]), | |
| for(v=1,25,if(!used[v], | |
| used[v]=1; | |
| if(isprime(ps[k]+ps[p]+ps[g]+ps[u]+ps[v]), | |
| for(w=1,25,if(!used[w], | |
| used[w]=1; | |
| if(isprime(ps[i]+ps[q]+ps[w]+ps[t]+ps[s]), | |
| for(x=1,25,if(!used[x], | |
| used[x]=1; | |
| if(isprime(ps[w]+ps[x]+ps[g]+ps[n]+ps[c]), | |
| for(y=1,25,if(!used[y], | |
| used[y]=1; | |
| if(isprime(ps[j]+ps[h]+ps[x]+ps[r]+ps[y]), | |
| if(isprime(ps[e]+ps[o]+ps[v]+ps[y]+ps[s]), | |
| print("{[",ps[a],",",ps[b],",",ps[c],",",ps[d],",",ps[e],";"); | |
| print(ps[l],",",ps[m],",",ps[n],",",ps[f],",",ps[o],";"); | |
| print(ps[k],",",ps[p],",",ps[g],",",ps[u],",",ps[v],";"); | |
| print(ps[j],",",ps[h],",",ps[x],",",ps[r],",",ps[y],";"); | |
| print(ps[i],",",ps[q],",",ps[w],",",ps[t],",",ps[s],"];}"); | |
| quit | |
| ); | |
| ); | |
| used[y]=0; | |
| )); | |
| ); | |
| used[x]=0; | |
| )); | |
| ); | |
| used[w]=0; | |
| )); | |
| ); | |
| used[v]=0; | |
| )); | |
| ); | |
| used[u]=0; | |
| )); | |
| used[t]=0; | |
| )); | |
| ); | |
| used[s]=0; | |
| )); | |
| used[r]=0; | |
| )); | |
| ); | |
| used[q]=0; | |
| )); | |
| used[p]=0; | |
| )); | |
| ); | |
| used[o]=0; | |
| )); | |
| used[n]=0; | |
| )); | |
| used[m]=0; | |
| )); | |
| ); | |
| used[l]=0; | |
| )); | |
| used[k]=0; | |
| )); | |
| used[j]=0; | |
| )); | |
| ); | |
| used[i]=0; | |
| )); | |
| used[h]=0; | |
| )); | |
| used[g]=0; | |
| )); | |
| used[f]=0; | |
| )); | |
| ); | |
| used[e]=0; | |
| )); | |
| used[d]=0; | |
| )); | |
| used[c]=0; | |
| )); | |
| used[b]=0; | |
| )); | |
| used[a]=0; | |
| )); | |
| } |
These are the possible sets fo 25 distinct primes summing up to 1171:
pi@raspberrypi5:~/PrimeSquares $ gp -q < sum25e.gp
205 primes
1171
[3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 113]
[3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 109]
[3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 103, 107]
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ cat sum25e.gp
min25=vecsum(primes(26))-2;
mx=1279-1;
P=Vecrev(primes([3,mx]));
print(#P," primes");
srch(s,v,i,n)={
if(s<0,return);
if(n==0,if(s==0,print(Vecrev(v));return));
if(i>#P,return);
if(#P-i+1<n,return);
if(vecsum(P[#P-n+1..#P])>s,return);
srch(s-P[i],concat(v,[P[i]]),i+1,n-1);
srch(s,v,i+1,n);
};
print(1171);
srch(1171,[],1,25);
pi@raspberrypi5:~/PrimeSquares $
The 1st vector is used in above gist.
Minimal total sum that is a prime of 25 distinct primes is 1171.
Above gist computes a solution superfast:
pi@raspberrypi5:~/PrimeSquares $ time gp -q < sum25f.gp
{[3,5,7,11,17;
67,41,43,13,47;
37,53,19,59,113;
31,23,71,61,97;
29,89,83,79,73];}
real 0m0.031s
user 0m0.028s
sys 0m0.004s
pi@raspberrypi5:~/PrimeSquares $
Validation:
pi@raspberrypi5:~/PrimeSquares $ cat > 1171.gp
{[3,5,7,11,17;
67,41,43,13,47;
37,53,19,59,113;
31,23,71,61,97;
29,89,83,79,73];}
pi@raspberrypi5:~/PrimeSquares $ ps=1171.gp gp -q < validate.gp
1171.gp, sum: 1171, validated
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ cat validate.gp
assert(b)={if(!(b),error("assertion failed"));}
{
ps=eval(readvec(getenv("ps"))[1]);
print1(getenv("ps"),", ");
s=#ps;
assert(#ps~==s);
foreach(ps,c,assert(#c==s));
foreach(ps~,r,assert(#r==s));
foreach(ps,c,assert(isprime(vecsum(c))));
foreach(ps~,r,assert(isprime(vecsum(r))));
assert(isprime(trace(ps)));
assert(isprime(vecsum([ps[i,#ps+1-i]|i<-[1..#ps]])));
S=[r|i<-[1..s];r<-ps[i,]];
assert(#Set(S)==s^2);
foreach(S,e,assert(isprime(e)));
assert(isprime(vecsum(S)));
print("sum: ",vecsum(S),", validated");
}
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ gp -q < sum25d.gp
205 primes
1163: 0
1171: 3
1181: 2
1187: 2
1193: 5
1201: 19
1213: 38
1217: 27
1223: 39
1229: 65
1231: 102
1237: 140
1249: 263
1259: 306
1277: 695
sum=1706
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ cat sum25d.gp
min25=vecsum(primes(26))-2;
mx=1279-1;
P=Vecrev(primes([3,mx]));
print(#P," primes");
cnt=0;
srch(s,v,i,n)={
if(s<0,return);
if(n==0,if(s==0,cnt+=1;return));\\print(v);cnt+=1;return));
if(i>#P,return);
if(#P-i+1<n,return);
if(vecsum(P[#P-n+1..#P])>s,return);
srch(s-P[i],concat(v,[P[i]]),i+1,n-1);
srch(s,v,i+1,n);
};
{
C=primes([min25,mx]);
Sum=0;
foreach(C,c,print1(c,": ");cnt=0;srch(c,[],1,25);print(cnt);Sum+=cnt);
print("sum=",Sum);
}
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ gp -q < sum25c.gp
205 primes
[1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277]
M[205,1163]=[1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23]
[1159, 1161, 1165, 1167, 1169, 1171, 1173, 1175, 1177, 1179, 1181, 1183, 1185, 1187, 1189, 1191, 1193, 1195, 1197, 1199, 1201, 1203, 1205, 1207, 1209, 1211, 1213, 1215, 1217, 1219, 1221, 1223, 1225, 1227, 1229, 1231, 1233, 1235, 1237, 1239, 1241, 1243, 1245, 1247, 1249, 1251, 1253, 1255, 1257, 1259, 1261, 1263, 1265, 1267, 1269, 1271, 1273, 1275, 1277]
pi@raspberrypi5:~/PrimeSquares $
pi@raspberrypi5:~/PrimeSquares $ cat sum25c.gp
{
min25=vecsum(primes(26))-2;
mx=1279-1;
P=primes([5,mx]);
print(#P+1," primes");
M=matrix(1+#P,mx,y,x,Set());
y=1;M[y,3]=Set([1]);
foreach(P,p,
y+=1;
M[y,p]=Set([1]);
for(x=1,mx,M[y,x]=setunion(M[y-1,x],M[y,x]));
for(x=1,mx-p,M[y,x+p]=setunion(Set([e+1|e<-M[y-1,x]]),M[y,x+p]));
);
C=[x|x<-primes([min25,mx]),setsearch(M[y,x],25)];
print(C);
print("M[205,1163]=",M[y,1163]);
print("\n",[x|x<-[1..mx],setsearch(M[y,x],25)]);
}
pi@raspberrypi5:~/PrimeSquares $
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Smallest total sum prime 5x5 square of distinct primes has sum 1171:
https://www.mersenneforum.org/node/1055343?p=1062055#post1062055