mod x^4+1

[paulunderwood]

You may be aware that no one has found a counterexample to the test Mod(Mod(x+2,n),x^2+1)^(n+1)==5 for n%4==3.

I propose the test Mod(Mod(x+1,n),x^4+1)^(n-1) for odd n with the following results

• n%8==1: 1: counterexamples exist.
• n%8==3: x^2-x+1
• n%8==5: -x^3+x^2-x
• n%8==7: -x^3

[Dr Sardonicus]

Originally Posted by paulunderwood

You may be aware that no one has found a counterexample to the test Mod(Mod(x+2,n),x^2+1)^(n+1)==5 for n%4==3.

I propose the test Mod(Mod(x+1,n),x^4+1)^(n-1) for odd n with the following results

• n%8==1: 1: counterexamples exist.
• n%8==3: x^2-x+1
• n%8==5: -x^3+x^2-x
• n%8==7: -x^3

I was able to prove that the indicated congruences indeed hold if n is a prime congruent to 3, 5, or 7 (mod 8). It made an interesting exercise.

So there are no counterexamples to these with n prime

I have no idea how to try to construct composite n congruent to 3, 5, or 7 (mod 8) for which these congruences hold.

EDIT: BTW, the test for numbers congruent to 1 (mod 4) corresponding to the test you gave for numbers congruent to 3 (mod 4) is

lift(lift(Mod(Mod(1,n)*x + 2,x^2 + 1)^(n-1))) == 1.

Every prime congruent to 1 (mod 4) other than 5 satisfies this condition. (Why 5 is exceptional is left as an exercise for the reader.)

But the condition is rife with pseudoprimes, as the following mindless sweep shows:

? forstep(n=5,100000,4,r=lift(lift(Mod(Mod(1,n)*x+2,x^2+1)^(n-1)));if(r==1&&!ispseudoprime(n),print(n" "factor(n))))
15841 [7, 1; 31, 1; 73, 1]
29341 [13, 1; 37, 1; 61, 1]
38081 [113, 1; 337, 1]
40501 [101, 1; 401, 1]
41041 [7, 1; 11, 1; 13, 1; 41, 1]
46657 [13, 1; 37, 1; 97, 1]
75361 [11, 1; 13, 1; 17, 1; 31, 1]

For the mod x^4 + 1 test for numbers congruent to 1 (mod 8), we have

? forstep(n=9,500000,8,r=lift(lift(Mod(Mod(1,n)*x+1,x^4+1)^(n-1)));if(r==1&&!ispseudoprime(n),print(n" "factor(n))))
15841 [7, 1; 31, 1; 73, 1]
162401 [17, 1; 41, 1; 233, 1]
410041 [41, 1; 73, 1; 137, 1]

It occurred to me that a test of numbers congruent to 1 (mod m) based on the cyclotomic polynomial for the primitive m-th roots of unity will also be rife with pseudoprimes. The reason is that it is well known that there are infinitely many Carmichael numbers whose factors are all congruent to 1 (mod m). Any such Carmichael number (with the possible exceptions of those divisible by a finite set of primes) will be a pseudoprime for such a test. Note, however, that although 15841 is a Carmichael number, not all its factors are congruent to 1 (mod 4). And two of the pseudoprimes for the mod x^2 + 1 test only have two prime factors, so are automatically not Carmichael numbers.

[paulunderwood]

The test over x^2+1 can be computed with 2 Selfridges. Over x^4+1 it is 8 Selfridges by combining the difference of squares:

? Mod(s*x^3+t*x^2+u*x+v,x^4+1)^2                           
Mod((2*v*s + 2*u*t)*x^3 + (-s^2 + (2*v*t + u^2))*x^2 + (-2*t*s + 2*v*u)*x + (-2*u*s + (-t^2 + v^2)), x^4 + 1)         
? Mod(s*x^3+t*x^2+u*x+v,x^4+1)*(x+1)                       
Mod((s + t)*x^3 + (t + u)*x^2 + (u + v)*x + (-s + v), x^4 + 1)

where s,t,u and v are intermediate values during left-right binary exponentiation. (x+1) is the base.

[Dr Sardonicus]

Originally Posted by paulunderwood

You may be aware that no one has found a counterexample to the test Mod(Mod(x+2,n),x^2+1)^(n+1)==5 for n%4==3.
<snip>

Just out of curiosity, up to what limit has this been checked?

[paulunderwood]

Originally Posted by Dr Sardonicus

Just out of curiosity, up to what limit has this been checked?

I have checked the n%3==4 test over x^2+1 up to 2^50.

The others I tend to test up to 10^8 -- takes a minute or two.

Furthermore, for n%4==3 I can test base x+2 over x^2+1, for n%8==5 base x+1 over x^4+1, for n%16==9 base x+1 over x^8+1 and so on, with the number of Selfridges ballooning.

[Dr Sardonicus]

Originally Posted by paulunderwood

I have checked the n%3==4 test over x^2+1 up to 2^50.

Hmm. Might be time to start thinking about how to construct composite n that "fool" the test.

The others I tend to test up to 10^8 -- takes a minute or two.

If you turn up pseudoprimes < 10^8, I'd say abandon that test.

Furthermore, for n%4==3 I can test base x+2 over x^2+1, for n%8==5 base x+1 over x^4+1, for n%16==9 base x+1 over x^8+1 and so on, with the number of Selfridges ballooning.

How many tests

lift(lift(Mod(Mod(1,n)*x,x^2 - a*x + 1)^(n+1)))==1

with a-values such that kronecker(a^2 - 4, n) == -1 could you run for the same number of Selfridges?

(I'm assuming the computational cost of determining such a-values is negligible in comparison. I am also disregarding any pertinent gcd(poly in a, n) conditions which I assume are likewise computationally very cheap.)

[paulunderwood]

Originally Posted by Dr Sardonicus

How many tests

lift(lift(Mod(Mod(1,n)*x,x^2 - a*x + 1)^(n+1)))==1

with a-values such that kronecker(a^2 - 4, n) == -1 could you run for the same number of Selfridges?

(I'm assuming the computational cost of determining such a-values is negligible in comparison. I am also disregarding any pertinent gcd(poly in a, n) conditions which I assume are likewise computationally very cheap.)

That is easy for testing x+1 over x^4+1; It is 8/2=4 times the number of Selfridges of x over x^2-a*x+1 (with (a^2-4 | n)==-1). I have not computed the function for x^2^k+1, but it gets big!

[paulunderwood]

Considering n%8==5 with Mod(Mod(x+1,n),x^4+1)^n==1-x as the main test, can you find any k such that Mod(Mod(x+1,n),x^4+1)^k==1-x for composite n?

[Dr Sardonicus]

Originally Posted by paulunderwood

Considering n%8==5 with Mod(Mod(x+1,n),x^4+1)^n==1-x as the main test, can you find any k such that Mod(Mod(x+1,n),x^4+1)^k==1-x for composite n?

(checks, OK, it's equivalent to test proposed earlier in thread).

Evidently, none up to 10^8 since you've tested that far. I don't intend to do a numerical sweep above that point, so if you want more searching above 10^8, you'll have to do it yourself or get someone else to do it.

I have been thinking about ways to construct pseudoprimes - but no good ideas yet.

[paulunderwood]

Originally Posted by Dr Sardonicus

(checks, OK, it's equivalent to test proposed earlier in thread).

Evidently, none up to 10^8 since you've tested that far. I don't intend to do a numerical sweep above that point, so if you want more searching above 10^8, you'll have to do it yourself or get someone else to do it.

I have been thinking about ways to construct pseudoprimes - but no good ideas yet.

I did say "any k". Powers k never seem to map x+1 to 1-x. HTH.

[paulunderwood]

Originally Posted by Dr Sardonicus

If you turn up pseudoprimes < 10^8, I'd say abandon that test.

n%8==5 tested up to n<10^10. I will take it to the next exponent.