Abstract text (incl. figure legends and references)

Background
Significant interests have been generated in the past years in applying machine learning techniques to headaches diagnosis. Here we propose that headaches disorder diagnosis using ICHD3 can be translated to a prime factorization problem.
Methods
We assigned each statement of headache characteristic in the ICHD3 criteria to a unique prime number. (Table 1). Each ICHD3 diagnosis criterion can be decomposed to logic statements, which then in turn can be used to generate a list of composite numbers (Table 2) through the following rules: 1) If the logic AND is required, we multiply the two numbers. 2) If the logic OR is required, we document both numbers in a list.
To use the algorithm, the user answers yes or no to each of the headache characteristics in Table 1; then the user raises the corresponding prime numbers to 1st power if the answer is yes and 0th power if the answer is no. These numbers are then multiplied together; if this number can be divided by a number in Table 2 without remainder, then the corresponding ICHD3 diagnosis is diagnostic.
For example, a patient with greater than 5 episodes (19) of pulsating (71), unilateral headaches (73) lasting 5 hours each (17) with nausea (37) but no photo/phonophobia (43*47) can be represented by a score of 19*17*37*43*47*71*73 =
125184848693. This number can be divided by 61942033 without remainder. Since 61942033 corresponds to migraine without aura (Table 2), the diagnosis is therefore migraine without aura.
Results
We applied the above method to migraine without aura as well as infrequent, frequent, and chronic tension type headaches as a pilot study. For migraine without aura, infrequent, frequent, and chronic tension type headaches, 22 composite numbers represent each of the diagnostic criteria.
Conclusion
Applying headache diagnosis according to the ICHD3 criteria can be translated to a prime factorization problem. Our method represents a bridge between number theory and clinical medicine.