A Formula for Composite Numbers

Here we state a theorem that finds composite numbers. Proof will be given elsewhere.

Theorem:

Let and

Define for in and in .

If , then is a composite number.

A = [a for a in range(1,11)] and B = [b for b in range(1,11)]

N = [0.5*(a+b+1+ math.sqrt((a-b)**2+2*a+2*b+5)) for a in A for b in B] print(“N=”,sorted(N))

In the data below the only integers are X = {3, 5, 7, 8, 9, 11, …}. In this set the missing number plus one is prime. Whereas if x in X, then x+1 is composite.

N= [3.0, 3.732050807568877, 3.732050807568877, 4.302775637731995, 4.561552812808831, 4.561552812808831, 5.0, 5.0, 5.449489742783178, 5.449489742783178, 5.561552812808831, 5.7912878474779195, 5.7912878474779195, 6.23606797749979, 6.23606797749979, 6.372281323269014, 6.372281323269014, 6.645751311064591, 6.645751311064591, 6.7912878474779195, 7.0, 7.0, 7.3166247903554, 7.3166247903554, 7.449489742783178, 7.449489742783178, 7.541381265149109, 7.541381265149109, 7.82842712474619, 7.82842712474619, 8.0, 8.192582403567252, 8.192582403567252, 8.274917217635375, 8.274917217635375, 8.464101615137753, 8.464101615137753, 8.64575131106459, 8.64575131106459, 8.701562118716424, 8.701562118716424, 9.0, 9.0, 9.192582403567252, 9.242640687119284, 9.242640687119284, 9.372281323269014, 9.372281323269014, 9.405124837953327, 9.405124837953327, 9.60555127546399, 9.60555127546399, 9.82842712474619, 9.82842712474619, 9.854101966249685, 9.854101966249685, 10.16227766016838, 10.16227766016838, 10.2169905660283, 10.2169905660283, 10.358898943540673, 10.358898943540673, 10.372281323269014, 10.531128874149275, 10.531128874149275, 10.54138126514911, 10.54138126514911, 10.74165738677394, 10.74165738677394, 11.0, 11.0, 11.0, 11.0, 11.196152422706632, 11.196152422706632, 11.3166247903554, 11.3166247903554, 11.321825380496477, 11.321825380496477, 11.47213595499958, 11.47213595499958, 11.54138126514911, 11.653311931459037, 11.653311931459037, 11.701562118716424, 11.701562118716424, 11.872983346207416, 11.872983346207416, 12.140054944640259, 12.140054944640259, 12.16227766016838, 12.16227766016838, 12.464101615137753, 12.464101615137753, 12.701562118716424, 12.854101966249685, 12.854101966249685, 13.3166247903554, 13.3166247903554, 13.854101966249685] ​