This memo explores the properties and distribution of prime numbers representable by the quadratic form $2x^2 + xy + 3y^2$. Utilizing an optimized primality testing method based on matrix exponentiation, we efficiently identify such primes, even extending to 300-digit numbers. The approach hinges on the irreducibility of the polynomial $X^3 – X – 1$ modulo a prime $p$, correlating with the representability by the given quadratic form. This method aligns perfectly with the OEIS sequence A106867, which catalogs primes satisfying specific algebraic conditions.
Our implementation leverages a third-order linear recurrence relation derived from the polynomial, optimized via matrix exponentiation to reduce computational complexity from linear to logarithmic time. This optimization enables rapid verification of primes within extensive numerical ranges. Comparative analysis with existing OEIS recipes demonstrates our method’s superior efficiency and scalability.
Preliminary results indicate consistent identification of primes within the tested range, with no false positives detected among composites. The integration of the Ramanujan tau function further enhances the test’s discriminative power, providing additional congruence conditions that bolster accuracy. This synergy between algebraic number theory and computational techniques underscores the robustness of our approach.
The implications of this work extend to various domains, including cryptography and computational mathematics, where efficient prime verification is paramount. Future endeavors will focus on extending the methodology to other quadratic forms and refining the algorithm to accommodate even larger primes. Additionally, theoretical investigations into the underlying algebraic structures may yield deeper insights into prime distributions and their representations.
In conclusion, the development of an optimized primality test for primes of the form $2x^2 + xy + 3y^2$ represents a significant advancement in computational number theory. Its alignment with established sequences and demonstrated efficiency paves the way for further research and practical applications in related fields. [ Note: What the AI wrote isn’t that great, but it’s in Latex and it’s a passable first draft] Nov 2, 3pm ]
? for(U=10000000,10001000,if(isfrob3prime(U),printf("%d ",U)))
10000103 10000229 10000247 10000271 10000303 10000339 10000349 10000363 10000379 10000481 10000609 10000657 10000723 10000763 10000799 10000873 10000891 10000961 10000987 10000993 // isfrob3prime is a cubic Frobenius-based probable prime test
// for the quadratic form in the title
? for(U=10000000,10001000,if(isqfrep(U),printf("%d ",U)))
10000103 10000229 10000247 10000271 10000303 10000339 10000349 10000363 10000379 10000481 10000609 10000657 10000723 10000763 10000799 10000873 10000891 10000961 10000987 10000993 // issqfrep is based on the comments to OEIS sequence A106867
Here is the code needed for isfrob3prime:
? isfrob3prime
%967 = (ZZZ)->p=ZZZ;init(p);Mpp=power(M,p);output=Mpp*v3;test=((output[2,1]^3-output[2,1]-Mod(24,p)))==Mod(0,p);test=test&&(((output[3,1]+Mod(1,p)))==Mod(0,p));test=test&&(((output[1,1]+Mod(1,p)))==Mod(0,p));return(test)
? init
%968 = (p)->M[1,1]=Mod(0,p);M[1,2]=Mod(1,p);M[1,3]=Mod(1,p);M[2,1]=Mod(1,p);M[2,2]=M[1,1];M[2,3]=M[1,1];M[3,1]=M[1,1];M[3,2]=M[1,2];M[3,3]=M[1,1];v3[1,1]=Mod(3,p);v3[2,1]=Mod(2,p);v3[3,1]=Mod(0,p)
? power
%969 = (a,n)->v=digits(n,2);l=length(v);curpow=a;for(Y=2,l,ap2=curpow*curpow;if(v[Y],ap2=a*ap2);curpow=ap2;);return(curpow)
Here is the code for isqfrep:
isqfrep =
(n)->val=0;if(isprime(n),if((ramanujantau(n)%23)==22,val=1));return(val)
