Is this a three-digit number ending in one?
JavaScript (ES7), 43 40 39 bytes

f=(n,b)=&gt;n&lt;b*b?0:n%b==1&amp;n&lt;b**3|f(n,-~b)


Test cases



f=(n,b)=&gt;n&lt;b*b?0:n%b==1&amp;n&lt;b**3|f(n,-~b)

console.log('[Truthy]')
console.log(f(5))
console.log(f(73))
console.log(f(101))
console.log(f(1073))
console.log(f(17))
console.log(f(22))
console.log(f(36))
console.log(f(55))
console.log(f(99))
console.log(f(46656))
console.log(f(46657))
console.log(f(46658))
console.log(f(46659))
console.log(f(46660))
console.log(f(46663))
console.log(f(46665))
console.log(f(46666))
console.log(f(46670))
console.log(f(46671))

console.log('[Falsy]')
console.log(f(8))
console.log(f(18))
console.log(f(23))
console.log(f(27))
console.log(f(98))
console.log(f(90))
console.log(f(88))
console.log(f(72))
console.log(f(68))
console.log(f(46661))
console.log(f(46662))
console.log(f(46664))
console.log(f(46667))
console.log(f(46668))
console.log(f(46669))




Commented

f = (n,           // given n = input
        b) =&gt;     // and using b = base, initially undefined
  n &lt; b * b ?     // if n is less than b²:
    0             //   n has less than 3 digits in base b or above -&gt; failure
  :               // else:
    n % b == 1 &amp;  //   return a truthy value if n is congruent to 1 modulo b
    n &lt; b**3 |    //   and n is less than b³ (i.e. has less than 4 digits in base b)
    f(n, -~b)     //   or the above conditions are true for some greater value of b