Why is the minimum number of sprocket teeth 17?

Hello Mohamed!
Wheels can be made with few teeth (6, 5, etc) but they generate an undercut effect at the root of the tooth that weakens it. I am attaching two figures, one with normal toothing, in which the number of teeth generate strong profiles (with a wide base) and another figure in which you can see how a wheel with fewer teeth works, which requires reducing the thickness. from the base of the profile.
On the other hand, the reason an odd number of teeth is requested (11 or 17 in your case) is to get each tooth on the small gear to work "against every single" tooth on the other gear. When the amount is even, there is a situation where each tooth of the small wheel always meets the same teeth of the big wheel, and this can lead to uneven wear.
I hope the comment will help you.
Best regards!

Hello,
there is a lot of material on the internet if you search for "gear design" (videos, notes, books). What I am attaching is a short and interesting scientific article, of which you can look at the bibliographical references at the end and try to find one of the books it cites.
Best regards

This material is more extensive, of good origin and with bibliographical citations to deepen any aspect of the subject.

To add something visual to Bob's comments, I'm posting some pictures (from the SolidWorks toolbox) comparing a 9 tooth sprocket to a 25 tooth sprocket and a 50 tooth sprocket.
You can observe the geometric issues that lead to the weakening of the tooth when the amount is less and less...

Without intending to enter into a sterile discussion, the reasons why there is weakening of teeth (in the root when they are gears that mesh, and in the head when they are chain wheels) are quite similar: they are due to the kinematics of the gear (tooth -tooth in one case... and chain tooth in the other).

Of course there are also "technological/constructive issues" like the heat treatment ones cited by Bob, but I think the deeper reasons are purely kinematic... and lead to a reduction in resistant sections which makes the loads/transmissible torques less. when the number of teeth on any sprocket becomes smaller. It's a good exercise to draw a 4-tooth wheel (either involute or chain... and exaggerately observe this effect).

Best regards!

Sprockets are used with chain for power transmission. Their design is quite different from gears that mesh directly together with involute teeth.

Sprockets are available with less than 17 teeth. However they are typically limited to lower torque transmission for several reasons.

1. Sprocket teeth are flame case hardened. Small sprockets lack sufficient mass in the hub to self quench and achieve the hardness required. Through hardening could be done but at a much greater expense and would also result in a loss of toughness. Something sprockets require. Small sprockets are not hardened.

2. Due to geometric limitations it is not possible to transmit full torque on each tooth with a small number of teeth. The tangent angles of the chain links tends to accentuate the loads and also require a tooth profile with less surface area. Plus you’re dealing with less teeth to share the load.

No hardness + higher loads + less tooth area + less teeth = very high wear.

Again, sprockets with less than 17 teeth ARE available. They just have limitations. I often use sprockets with as few as 12 teeth but they are only used for low speed, low load, applications.

Keep in mind some small series sprockets like #25 are rarely available with hardened teeth. This is due to the small size and thickness, regardless of the number of teeth, making it difficult to achieve hardening. #25 chain is not available with roller links so wear is a problem. #25 chain and smaller have severe torque limitations.