Is this a three-digit number ending in one? Given a nonnegative integer, return whether it is a three digit number ending in one, in any consistent integer base. In other words, the number needs to be represented in base-N, N being an integer greater than zero.
Rules

This is code-golf, so shortest answer wins.
Since unary behaves weirdly, the behavior with input 310 is undefined.
Standard loopholes are prohibited.

Examples
True:
5   
73  
101 
1073
17
22
36
55
99  

False:
8
18
23
27
98
90
88
72
68

A handful of large numbers:
46656 true
46657 true
46658 true
46659 true
46660 true
46661 false
46662 false
46663 true
46664 false
46665 true
46666 true
46667 false
46668 false
46669 false
46670 true
46671 true
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