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A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

@მამუკა ჯიბლაძე, the simplest and easiest example would be a slight strengthening of the Euclid Theorem:

THEOREM (Euclid++) For every natural number (i.e. positive integer) $\ n,\ $ the product of primes $\le\ n\ $ is $\ \ge\ n$.

REMARK 1   The Euclid++ Theorem is in a sense strictly stronger than Euclid's theorem. (The Euclid++ Theorem is only the first in a sequence of still stronger elementary theorem, but so far they are all weak).

REMARK 2   Euclid didn't exist (it's rather clear to historians). I am sure that the so-called Euclid's Theorem was actually discovered and proved by Eratosthenes.

A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

@მამუკა ჯიბლაძე, the simplest and easiest example would be a slight strengthening of the Euclid Theorem:

THEOREM (Euclid++) For every natural number (i.e. positive integer) $\ n,\ $ the product of primes $\le\ n\ $ is $\ \ge\ n$.

A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

@მამუკა ჯიბლაძე, the simplest and easiest example would be a slight strengthening of the Euclid Theorem:

THEOREM (Euclid++) For every natural number (i.e. positive integer) $\ n,\ $ the product of primes $\le\ n\ $ is $\ \ge\ n$.

A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

@მამუკა ჯიბლაძე, the simplest and easiest example would be a slight strengthening of the Euclid Theorem:

THEOREM (Euclid++) For every natural number (i.e. positive integer) $\ n,\ $ the product of primes $\le\ n\ $ is $\ \ge\ n$.

REMARK 1   The Euclid++ Theorem is in a sense strictly stronger than Euclid's theorem. (The Euclid++ Theorem is only the first in a sequence of still stronger elementary theorem, but so far they are all weak).

REMARK 2   Euclid didn't exist (it's rather clear to historians). I am sure that the so-called Euclid's Theorem was actually discovered and proved by Eratosthenes.

A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

A comment (rather than an Answer).

(I have elementary examples of an induction $n\rightarrow n+1$ over all integers but the theorem $T(n)$ is about the set of primes $\le n$).

@მამუკა ჯიბლაძე, the simplest and easiest example would be a slight strengthening of the Euclid Theorem:

THEOREM (Euclid++) For every natural number (i.e. positive integer) $\ n,\ $ the product of primes $\le\ n\ $ is $\ \ge\ n$.