# Stem Quiz #3

Time for another STEM Quiz!
This is a problem by Ramanujan which I found on the list of “Solved Problems” on Wikipedia’s page for the floor function. Not too difficult - I had myself convinced in an hour and had a semi-formal proof another hour or two later.

Prove: \lfloor{\sqrt{n}+\sqrt{n+1}}\rfloor= \lfloor{\sqrt{4n+2}}\rfloor for n\in\mathbb{N}.

Recall that:

• \mathbb{Z} is the set of integers.
• \mathbb{N} is the set of natural numbers, that is, positive integers.
• \mathbb{R} is the set of real numbers, including integers, rationals, analytic, and trancendental numbers, but not imaginary or complex numbers.
• v\in W means that v is an element (member) of the set W.
• \lfloor{x}\rfloor is the “floor function,” the greatest integer not larger than x (x\in\mathbb{R}) so that \lfloor{x}\rfloor\in \mathbb{Z}, and x-1\lt\lfloor{x}\rfloor\le x.

Edit: Hmm… LaTex seems to work OK in Chrome and Edge, but not Firefox.

Yes, fixed!

1 Like

Digging your LaTex work, Gus. Takes me back to grad school, which is a long journey in my case.

Is there a question in there somewhere?

1 Like

The “question” is to:

Prove: \lfloor{\sqrt{n}+\sqrt{n+1}}\rfloor =\lfloor{\sqrt{4n+2}}\rfloor for n\in\mathbb{N}.

OBTW, while I knew LaTeX was a thing mid '90’s (well after my grad school days*), I’m actually learning it 35 years later as I composed this question, and even more so as I prepare my version of the proof. I’m late to this party, but I’m still learning new tools as I approach retirement.

* I actually typed up my (1986) master’s thesis to be printed on a daisy wheel printer, which I then put into an IBM typewriter with a mathematical symbols printball and typed in a bunch more, then penned in things like integral symbols, which was a pain because my penmanship is…less than exemplary. It would certainly be even worse if I hadn’t found the SCA (Society for Creative Anachronisms) and taken a calligraphy course; before that course, I couldn’t even read my own notes much of the time. I can now read them most of the time.

1 Like

Did you mean?

Prove: \lfloor{\sqrt{n}+\sqrt{n+1}}\rfloor = \lfloor{\sqrt{4n+2}}\rfloor for n\in\mathbb{N}.

1 Like

Probably yea:


import math

for n in range(1,1000):
lhs = math.floor(math.sqrt(n) + math.sqrt(n+1))
rhs = math.floor(math.sqrt(4*n + 2))

if(lhs > rhs):
result = ">"
elif (lhs < rhs):
result = "<"
else:
result = "="

print("{} : {} {} {}".format(str(n), str(lhs), (result), str(rhs)))


1 : 2 = 2
2 : 3 = 3
3 : 3 = 3
4 : 4 = 4
5 : 4 = 4
6 : 5 = 5
7 : 5 = 5
8 : 5 = 5
9 : 6 = 6
10 : 6 = 6
11 : 6 = 6
12 : 7 = 7
13 : 7 = 7
14 : 7 = 7
15 : 7 = 7
16 : 8 = 8
17 : 8 = 8
18 : 8 = 8
19 : 8 = 8
20 : 9 = 9
21 : 9 = 9
22 : 9 = 9
23 : 9 = 9
24 : 9 = 9
25 : 10 = 10
26 : 10 = 10
27 : 10 = 10
28 : 10 = 10
29 : 10 = 10
30 : 11 = 11
31 : 11 = 11
32 : 11 = 11
33 : 11 = 11
34 : 11 = 11
35 : 11 = 11
36 : 12 = 12
37 : 12 = 12
38 : 12 = 12
39 : 12 = 12
40 : 12 = 12
41 : 12 = 12
42 : 13 = 13
43 : 13 = 13
44 : 13 = 13
45 : 13 = 13
46 : 13 = 13
47 : 13 = 13
48 : 13 = 13
49 : 14 = 14
50 : 14 = 14
51 : 14 = 14
52 : 14 = 14
53 : 14 = 14
54 : 14 = 14
55 : 14 = 14
56 : 15 = 15
57 : 15 = 15
58 : 15 = 15
59 : 15 = 15
60 : 15 = 15
61 : 15 = 15
62 : 15 = 15
63 : 15 = 15
64 : 16 = 16
65 : 16 = 16
66 : 16 = 16
67 : 16 = 16
68 : 16 = 16
69 : 16 = 16
70 : 16 = 16
71 : 16 = 16
72 : 17 = 17
73 : 17 = 17
74 : 17 = 17
75 : 17 = 17
76 : 17 = 17
77 : 17 = 17
78 : 17 = 17
79 : 17 = 17
80 : 17 = 17
81 : 18 = 18
82 : 18 = 18
83 : 18 = 18
84 : 18 = 18
85 : 18 = 18
86 : 18 = 18
87 : 18 = 18
88 : 18 = 18
89 : 18 = 18
90 : 19 = 19
91 : 19 = 19
92 : 19 = 19
93 : 19 = 19
94 : 19 = 19
95 : 19 = 19
96 : 19 = 19
97 : 19 = 19
98 : 19 = 19
99 : 19 = 19
100 : 20 = 20
101 : 20 = 20
102 : 20 = 20
103 : 20 = 20
104 : 20 = 20
105 : 20 = 20
106 : 20 = 20
107 : 20 = 20
108 : 20 = 20
109 : 20 = 20
110 : 21 = 21
111 : 21 = 21
112 : 21 = 21
113 : 21 = 21
114 : 21 = 21
115 : 21 = 21
116 : 21 = 21
117 : 21 = 21
118 : 21 = 21
119 : 21 = 21
120 : 21 = 21
121 : 22 = 22
122 : 22 = 22
123 : 22 = 22
124 : 22 = 22
125 : 22 = 22
126 : 22 = 22
127 : 22 = 22
128 : 22 = 22
129 : 22 = 22
130 : 22 = 22
131 : 22 = 22
132 : 23 = 23
133 : 23 = 23
134 : 23 = 23
135 : 23 = 23
136 : 23 = 23
137 : 23 = 23
138 : 23 = 23
139 : 23 = 23
140 : 23 = 23
141 : 23 = 23
142 : 23 = 23
143 : 23 = 23
144 : 24 = 24
145 : 24 = 24
146 : 24 = 24
147 : 24 = 24
148 : 24 = 24
149 : 24 = 24
150 : 24 = 24
151 : 24 = 24
152 : 24 = 24
153 : 24 = 24
154 : 24 = 24
155 : 24 = 24
156 : 25 = 25
157 : 25 = 25
158 : 25 = 25
159 : 25 = 25
160 : 25 = 25
161 : 25 = 25
162 : 25 = 25
163 : 25 = 25
164 : 25 = 25
165 : 25 = 25
166 : 25 = 25
167 : 25 = 25
168 : 25 = 25
169 : 26 = 26
170 : 26 = 26
171 : 26 = 26
172 : 26 = 26
173 : 26 = 26
174 : 26 = 26
175 : 26 = 26
176 : 26 = 26
177 : 26 = 26
178 : 26 = 26
179 : 26 = 26
180 : 26 = 26
181 : 26 = 26
182 : 27 = 27
183 : 27 = 27
184 : 27 = 27
185 : 27 = 27
186 : 27 = 27
187 : 27 = 27
188 : 27 = 27
189 : 27 = 27
190 : 27 = 27
191 : 27 = 27
192 : 27 = 27
193 : 27 = 27
194 : 27 = 27
195 : 27 = 27
196 : 28 = 28
197 : 28 = 28
198 : 28 = 28
199 : 28 = 28
200 : 28 = 28
201 : 28 = 28
202 : 28 = 28
203 : 28 = 28
204 : 28 = 28
205 : 28 = 28
206 : 28 = 28
207 : 28 = 28
208 : 28 = 28
209 : 28 = 28
210 : 29 = 29
211 : 29 = 29
212 : 29 = 29
213 : 29 = 29
214 : 29 = 29
215 : 29 = 29
216 : 29 = 29
217 : 29 = 29
218 : 29 = 29
219 : 29 = 29
220 : 29 = 29
221 : 29 = 29
222 : 29 = 29
223 : 29 = 29
224 : 29 = 29
225 : 30 = 30
226 : 30 = 30
227 : 30 = 30
228 : 30 = 30
229 : 30 = 30
230 : 30 = 30
231 : 30 = 30
232 : 30 = 30
233 : 30 = 30
234 : 30 = 30
235 : 30 = 30
236 : 30 = 30
237 : 30 = 30
238 : 30 = 30
239 : 30 = 30
240 : 31 = 31
241 : 31 = 31
242 : 31 = 31
243 : 31 = 31
244 : 31 = 31
245 : 31 = 31
246 : 31 = 31
247 : 31 = 31
248 : 31 = 31
249 : 31 = 31
250 : 31 = 31
251 : 31 = 31
252 : 31 = 31
253 : 31 = 31
254 : 31 = 31
255 : 31 = 31
256 : 32 = 32
257 : 32 = 32
258 : 32 = 32
259 : 32 = 32
260 : 32 = 32
261 : 32 = 32
262 : 32 = 32
263 : 32 = 32
264 : 32 = 32
265 : 32 = 32
266 : 32 = 32
267 : 32 = 32
268 : 32 = 32
269 : 32 = 32
270 : 32 = 32
271 : 32 = 32
272 : 33 = 33
273 : 33 = 33
274 : 33 = 33
275 : 33 = 33
276 : 33 = 33
277 : 33 = 33
278 : 33 = 33
279 : 33 = 33
280 : 33 = 33
281 : 33 = 33
282 : 33 = 33
283 : 33 = 33
284 : 33 = 33
285 : 33 = 33
286 : 33 = 33
287 : 33 = 33
288 : 33 = 33
289 : 34 = 34
290 : 34 = 34
291 : 34 = 34
292 : 34 = 34
293 : 34 = 34
294 : 34 = 34
295 : 34 = 34
296 : 34 = 34
297 : 34 = 34
298 : 34 = 34
299 : 34 = 34
300 : 34 = 34
301 : 34 = 34
302 : 34 = 34
303 : 34 = 34
304 : 34 = 34
305 : 34 = 34
306 : 35 = 35
307 : 35 = 35
308 : 35 = 35
309 : 35 = 35
310 : 35 = 35
311 : 35 = 35
312 : 35 = 35
313 : 35 = 35
314 : 35 = 35
315 : 35 = 35
316 : 35 = 35
317 : 35 = 35
318 : 35 = 35
319 : 35 = 35
320 : 35 = 35
321 : 35 = 35
322 : 35 = 35
323 : 35 = 35
324 : 36 = 36
325 : 36 = 36
326 : 36 = 36
327 : 36 = 36
328 : 36 = 36
329 : 36 = 36
330 : 36 = 36
331 : 36 = 36
332 : 36 = 36
333 : 36 = 36
334 : 36 = 36
335 : 36 = 36
336 : 36 = 36
337 : 36 = 36
338 : 36 = 36
339 : 36 = 36
340 : 36 = 36
341 : 36 = 36
342 : 37 = 37
343 : 37 = 37
344 : 37 = 37
345 : 37 = 37
346 : 37 = 37
347 : 37 = 37
348 : 37 = 37
349 : 37 = 37
350 : 37 = 37
351 : 37 = 37
352 : 37 = 37
353 : 37 = 37
354 : 37 = 37
355 : 37 = 37
356 : 37 = 37
357 : 37 = 37
358 : 37 = 37
359 : 37 = 37
360 : 37 = 37
361 : 38 = 38
362 : 38 = 38
363 : 38 = 38
364 : 38 = 38
365 : 38 = 38
366 : 38 = 38
367 : 38 = 38
368 : 38 = 38
369 : 38 = 38
370 : 38 = 38
371 : 38 = 38
372 : 38 = 38
373 : 38 = 38
374 : 38 = 38
375 : 38 = 38
376 : 38 = 38
377 : 38 = 38
378 : 38 = 38
379 : 38 = 38
380 : 39 = 39
381 : 39 = 39
382 : 39 = 39
383 : 39 = 39
384 : 39 = 39
385 : 39 = 39
386 : 39 = 39
387 : 39 = 39
388 : 39 = 39
389 : 39 = 39
390 : 39 = 39
391 : 39 = 39
392 : 39 = 39
393 : 39 = 39
394 : 39 = 39
395 : 39 = 39
396 : 39 = 39
397 : 39 = 39
398 : 39 = 39
399 : 39 = 39
400 : 40 = 40
401 : 40 = 40
402 : 40 = 40
403 : 40 = 40
404 : 40 = 40
405 : 40 = 40
406 : 40 = 40
407 : 40 = 40
408 : 40 = 40
409 : 40 = 40
410 : 40 = 40
411 : 40 = 40
412 : 40 = 40
413 : 40 = 40
414 : 40 = 40
415 : 40 = 40
416 : 40 = 40
417 : 40 = 40
418 : 40 = 40
419 : 40 = 40
420 : 41 = 41
421 : 41 = 41
422 : 41 = 41
423 : 41 = 41
424 : 41 = 41
425 : 41 = 41
426 : 41 = 41
427 : 41 = 41
428 : 41 = 41
429 : 41 = 41
430 : 41 = 41
431 : 41 = 41
432 : 41 = 41
433 : 41 = 41
434 : 41 = 41
435 : 41 = 41
436 : 41 = 41
437 : 41 = 41
438 : 41 = 41
439 : 41 = 41
440 : 41 = 41
441 : 42 = 42
442 : 42 = 42
443 : 42 = 42
444 : 42 = 42
445 : 42 = 42
446 : 42 = 42
447 : 42 = 42
448 : 42 = 42
449 : 42 = 42
450 : 42 = 42
451 : 42 = 42
452 : 42 = 42
453 : 42 = 42
454 : 42 = 42
455 : 42 = 42
456 : 42 = 42
457 : 42 = 42
458 : 42 = 42
459 : 42 = 42
460 : 42 = 42
461 : 42 = 42
462 : 43 = 43
463 : 43 = 43
464 : 43 = 43
465 : 43 = 43
466 : 43 = 43
467 : 43 = 43
468 : 43 = 43
469 : 43 = 43
470 : 43 = 43
471 : 43 = 43
472 : 43 = 43
473 : 43 = 43
474 : 43 = 43
475 : 43 = 43
476 : 43 = 43
477 : 43 = 43
478 : 43 = 43
479 : 43 = 43
480 : 43 = 43
481 : 43 = 43
482 : 43 = 43
483 : 43 = 43
484 : 44 = 44
485 : 44 = 44
486 : 44 = 44
487 : 44 = 44
488 : 44 = 44
489 : 44 = 44
490 : 44 = 44
491 : 44 = 44
492 : 44 = 44
493 : 44 = 44
494 : 44 = 44
495 : 44 = 44
496 : 44 = 44
497 : 44 = 44
498 : 44 = 44
499 : 44 = 44
500 : 44 = 44
501 : 44 = 44
502 : 44 = 44
503 : 44 = 44
504 : 44 = 44
505 : 44 = 44
506 : 45 = 45
507 : 45 = 45
508 : 45 = 45
509 : 45 = 45
510 : 45 = 45
511 : 45 = 45
512 : 45 = 45
513 : 45 = 45
514 : 45 = 45
515 : 45 = 45
516 : 45 = 45
517 : 45 = 45
518 : 45 = 45
519 : 45 = 45
520 : 45 = 45
521 : 45 = 45
522 : 45 = 45
523 : 45 = 45
524 : 45 = 45
525 : 45 = 45
526 : 45 = 45
527 : 45 = 45
528 : 45 = 45
529 : 46 = 46
530 : 46 = 46
531 : 46 = 46
532 : 46 = 46
533 : 46 = 46
534 : 46 = 46
535 : 46 = 46
536 : 46 = 46
537 : 46 = 46
538 : 46 = 46
539 : 46 = 46
540 : 46 = 46
541 : 46 = 46
542 : 46 = 46
543 : 46 = 46
544 : 46 = 46
545 : 46 = 46
546 : 46 = 46
547 : 46 = 46
548 : 46 = 46
549 : 46 = 46
550 : 46 = 46
551 : 46 = 46
552 : 47 = 47
553 : 47 = 47
554 : 47 = 47
555 : 47 = 47
556 : 47 = 47
557 : 47 = 47
558 : 47 = 47
559 : 47 = 47
560 : 47 = 47
561 : 47 = 47
562 : 47 = 47
563 : 47 = 47
564 : 47 = 47
565 : 47 = 47
566 : 47 = 47
567 : 47 = 47
568 : 47 = 47
569 : 47 = 47
570 : 47 = 47
571 : 47 = 47
572 : 47 = 47
573 : 47 = 47
574 : 47 = 47
575 : 47 = 47
576 : 48 = 48
577 : 48 = 48
578 : 48 = 48
579 : 48 = 48
580 : 48 = 48
581 : 48 = 48
582 : 48 = 48
583 : 48 = 48
584 : 48 = 48
585 : 48 = 48
586 : 48 = 48
587 : 48 = 48
588 : 48 = 48
589 : 48 = 48
590 : 48 = 48
591 : 48 = 48
592 : 48 = 48
593 : 48 = 48
594 : 48 = 48
595 : 48 = 48
596 : 48 = 48
597 : 48 = 48
598 : 48 = 48
599 : 48 = 48
600 : 49 = 49
601 : 49 = 49
602 : 49 = 49
603 : 49 = 49
604 : 49 = 49
605 : 49 = 49
606 : 49 = 49
607 : 49 = 49
608 : 49 = 49
609 : 49 = 49
610 : 49 = 49
611 : 49 = 49
612 : 49 = 49
613 : 49 = 49
614 : 49 = 49
615 : 49 = 49
616 : 49 = 49
617 : 49 = 49
618 : 49 = 49
619 : 49 = 49
620 : 49 = 49
621 : 49 = 49
622 : 49 = 49
623 : 49 = 49
624 : 49 = 49
625 : 50 = 50
626 : 50 = 50
627 : 50 = 50
628 : 50 = 50
629 : 50 = 50
630 : 50 = 50
631 : 50 = 50
632 : 50 = 50
633 : 50 = 50
634 : 50 = 50
635 : 50 = 50
636 : 50 = 50
637 : 50 = 50
638 : 50 = 50
639 : 50 = 50
640 : 50 = 50
641 : 50 = 50
642 : 50 = 50
643 : 50 = 50
644 : 50 = 50
645 : 50 = 50
646 : 50 = 50
647 : 50 = 50
648 : 50 = 50
649 : 50 = 50
650 : 51 = 51
651 : 51 = 51
652 : 51 = 51
653 : 51 = 51
654 : 51 = 51
655 : 51 = 51
656 : 51 = 51
657 : 51 = 51
658 : 51 = 51
659 : 51 = 51
660 : 51 = 51
661 : 51 = 51
662 : 51 = 51
663 : 51 = 51
664 : 51 = 51
665 : 51 = 51
666 : 51 = 51
667 : 51 = 51
668 : 51 = 51
669 : 51 = 51
670 : 51 = 51
671 : 51 = 51
672 : 51 = 51
673 : 51 = 51
674 : 51 = 51
675 : 51 = 51
676 : 52 = 52
677 : 52 = 52
678 : 52 = 52
679 : 52 = 52
680 : 52 = 52
681 : 52 = 52
682 : 52 = 52
683 : 52 = 52
684 : 52 = 52
685 : 52 = 52
686 : 52 = 52
687 : 52 = 52
688 : 52 = 52
689 : 52 = 52
690 : 52 = 52
691 : 52 = 52
692 : 52 = 52
693 : 52 = 52
694 : 52 = 52
695 : 52 = 52
696 : 52 = 52
697 : 52 = 52
698 : 52 = 52
699 : 52 = 52
700 : 52 = 52
701 : 52 = 52
702 : 53 = 53
703 : 53 = 53
704 : 53 = 53
705 : 53 = 53
706 : 53 = 53
707 : 53 = 53
708 : 53 = 53
709 : 53 = 53
710 : 53 = 53
711 : 53 = 53
712 : 53 = 53
713 : 53 = 53
714 : 53 = 53
715 : 53 = 53
716 : 53 = 53
717 : 53 = 53
718 : 53 = 53
719 : 53 = 53
720 : 53 = 53
721 : 53 = 53
722 : 53 = 53
723 : 53 = 53
724 : 53 = 53
725 : 53 = 53
726 : 53 = 53
727 : 53 = 53
728 : 53 = 53
729 : 54 = 54
730 : 54 = 54
731 : 54 = 54
732 : 54 = 54
733 : 54 = 54
734 : 54 = 54
735 : 54 = 54
736 : 54 = 54
737 : 54 = 54
738 : 54 = 54
739 : 54 = 54
740 : 54 = 54
741 : 54 = 54
742 : 54 = 54
743 : 54 = 54
744 : 54 = 54
745 : 54 = 54
746 : 54 = 54
747 : 54 = 54
748 : 54 = 54
749 : 54 = 54
750 : 54 = 54
751 : 54 = 54
752 : 54 = 54
753 : 54 = 54
754 : 54 = 54
755 : 54 = 54
756 : 55 = 55
757 : 55 = 55
758 : 55 = 55
759 : 55 = 55
760 : 55 = 55
761 : 55 = 55
762 : 55 = 55
763 : 55 = 55
764 : 55 = 55
765 : 55 = 55
766 : 55 = 55
767 : 55 = 55
768 : 55 = 55
769 : 55 = 55
770 : 55 = 55
771 : 55 = 55
772 : 55 = 55
773 : 55 = 55
774 : 55 = 55
775 : 55 = 55
776 : 55 = 55
777 : 55 = 55
778 : 55 = 55
779 : 55 = 55
780 : 55 = 55
781 : 55 = 55
782 : 55 = 55
783 : 55 = 55
784 : 56 = 56
785 : 56 = 56
786 : 56 = 56
787 : 56 = 56
788 : 56 = 56
789 : 56 = 56
790 : 56 = 56
791 : 56 = 56
792 : 56 = 56
793 : 56 = 56
794 : 56 = 56
795 : 56 = 56
796 : 56 = 56
797 : 56 = 56
798 : 56 = 56
799 : 56 = 56
800 : 56 = 56
801 : 56 = 56
802 : 56 = 56
803 : 56 = 56
804 : 56 = 56
805 : 56 = 56
806 : 56 = 56
807 : 56 = 56
808 : 56 = 56
809 : 56 = 56
810 : 56 = 56
811 : 56 = 56
812 : 57 = 57
813 : 57 = 57
814 : 57 = 57
815 : 57 = 57
816 : 57 = 57
817 : 57 = 57
818 : 57 = 57
819 : 57 = 57
820 : 57 = 57
821 : 57 = 57
822 : 57 = 57
823 : 57 = 57
824 : 57 = 57
825 : 57 = 57
826 : 57 = 57
827 : 57 = 57
828 : 57 = 57
829 : 57 = 57
830 : 57 = 57
831 : 57 = 57
832 : 57 = 57
833 : 57 = 57
834 : 57 = 57
835 : 57 = 57
836 : 57 = 57
837 : 57 = 57
838 : 57 = 57
839 : 57 = 57
840 : 57 = 57
841 : 58 = 58
842 : 58 = 58
843 : 58 = 58
844 : 58 = 58
845 : 58 = 58
846 : 58 = 58
847 : 58 = 58
848 : 58 = 58
849 : 58 = 58
850 : 58 = 58
851 : 58 = 58
852 : 58 = 58
853 : 58 = 58
854 : 58 = 58
855 : 58 = 58
856 : 58 = 58
857 : 58 = 58
858 : 58 = 58
859 : 58 = 58
860 : 58 = 58
861 : 58 = 58
862 : 58 = 58
863 : 58 = 58
864 : 58 = 58
865 : 58 = 58
866 : 58 = 58
867 : 58 = 58
868 : 58 = 58
869 : 58 = 58
870 : 59 = 59
871 : 59 = 59
872 : 59 = 59
873 : 59 = 59
874 : 59 = 59
875 : 59 = 59
876 : 59 = 59
877 : 59 = 59
878 : 59 = 59
879 : 59 = 59
880 : 59 = 59
881 : 59 = 59
882 : 59 = 59
883 : 59 = 59
884 : 59 = 59
885 : 59 = 59
886 : 59 = 59
887 : 59 = 59
888 : 59 = 59
889 : 59 = 59
890 : 59 = 59
891 : 59 = 59
892 : 59 = 59
893 : 59 = 59
894 : 59 = 59
895 : 59 = 59
896 : 59 = 59
897 : 59 = 59
898 : 59 = 59
899 : 59 = 59
900 : 60 = 60
901 : 60 = 60
902 : 60 = 60
903 : 60 = 60
904 : 60 = 60
905 : 60 = 60
906 : 60 = 60
907 : 60 = 60
908 : 60 = 60
909 : 60 = 60
910 : 60 = 60
911 : 60 = 60
912 : 60 = 60
913 : 60 = 60
914 : 60 = 60
915 : 60 = 60
916 : 60 = 60
917 : 60 = 60
918 : 60 = 60
919 : 60 = 60
920 : 60 = 60
921 : 60 = 60
922 : 60 = 60
923 : 60 = 60
924 : 60 = 60
925 : 60 = 60
926 : 60 = 60
927 : 60 = 60
928 : 60 = 60
929 : 60 = 60
930 : 61 = 61
931 : 61 = 61
932 : 61 = 61
933 : 61 = 61
934 : 61 = 61
935 : 61 = 61
936 : 61 = 61
937 : 61 = 61
938 : 61 = 61
939 : 61 = 61
940 : 61 = 61
941 : 61 = 61
942 : 61 = 61
943 : 61 = 61
944 : 61 = 61
945 : 61 = 61
946 : 61 = 61
947 : 61 = 61
948 : 61 = 61
949 : 61 = 61
950 : 61 = 61
951 : 61 = 61
952 : 61 = 61
953 : 61 = 61
954 : 61 = 61
955 : 61 = 61
956 : 61 = 61
957 : 61 = 61
958 : 61 = 61
959 : 61 = 61
960 : 61 = 61
961 : 62 = 62
962 : 62 = 62
963 : 62 = 62
964 : 62 = 62
965 : 62 = 62
966 : 62 = 62
967 : 62 = 62
968 : 62 = 62
969 : 62 = 62
970 : 62 = 62
971 : 62 = 62
972 : 62 = 62
973 : 62 = 62
974 : 62 = 62
975 : 62 = 62
976 : 62 = 62
977 : 62 = 62
978 : 62 = 62
979 : 62 = 62
980 : 62 = 62
981 : 62 = 62
982 : 62 = 62
983 : 62 = 62
984 : 62 = 62
985 : 62 = 62
986 : 62 = 62
987 : 62 = 62
988 : 62 = 62
989 : 62 = 62
990 : 62 = 62
991 : 62 = 62
992 : 63 = 63
993 : 63 = 63
994 : 63 = 63
995 : 63 = 63
996 : 63 = 63
997 : 63 = 63
998 : 63 = 63
999 : 63 = 63


1 Like

I mean, I’m old fashioned and did it in my head, but yeah. 4 Likes

Problem: Prove \lfloor\sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor for n\in\mathbb{N}.

Solution (un-tidy but with motivation):

The motivation here is that square roots, inequalities, and halves (note that n+n+1=2n+1=1/2(4n+2); technically wrong, but that’s the intuition) lead to AM-GM. AM-GM states that the arithmetic mean is greater than or equal to the geometric mean. For two-variables, that means \tfrac{a+b}{2}\geq\sqrt{ab}. With AM-GM in mind, we proceed.

We have square roots, so we ignore the floor condition for now and square, then manipulate:

\begin{align*}
(\sqrt{n}+\sqrt{n+1})^2&=n+n+1+2\sqrt{n}\sqrt{n+1} \
&=2n+1+2\sqrt{n(n+1)} \
\intertext{We have a sqrt(4n+2), and 2n+1 is half of that, so lets continue as follows:}
&=4n+2-(2n+1)+2\sqrt{n(n+1)} \
\intertext{the following steps are for procuring the AM-GM we seek}
&=4n+2-2\left(n+\frac{1}{2}+\sqrt{n(n+1)}\right) \
&=4n+2-2\left(\frac{2n}{2}+\frac{1}{2}+\sqrt{n(n+1)}\right) \
&=4n+2-2\left(\frac{2n+1}{2}+\sqrt{n(n+1)}\right) \
&=4n+2-2\left(\frac{n+(n+1)}{2}+\sqrt{n(n+1)}\right) \
\end{align*}

Woah, that’s looking interesting! Note that AM-GM states that [\frac{n+(n+1)}{2}\geq\sqrt{n(n+1)}.] This is a bit unsatisfactory. Let’s think about what we’re trying to accomplish here.

We desire for \lfloor\sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor. That is to say, you can’t fit an integer between \sqrt{n}+\sqrt{n+1} and \sqrt{4n+2}. One step further and we have: not perfect square fits between \left(\sqrt{n}+\sqrt{n+1}\right)^2 and \sqrt{4n+2}^2=4n+2.

Looking at our equation [(\sqrt{n}+\sqrt{n+1})^2=4n+2-2\left(\frac{n+(n+1)}{2}+\sqrt{n(n+1)}\right),] it would seem that we would have to prove that 2\left(\frac{n+(n+1)}{2}+\sqrt{n(n+1)}\right) is a small quantity.

With this in mind, note that [n+\frac{1}{2}=\frac{n+(n+1)}{2}\geq\sqrt{n(n+1)}>\sqrt{n^2}=n\text{ by AM-GM}.]

So, \tfrac{n+(n+1)}{2}-\sqrt{n(n+1)}\geq 0. Also, because \sqrt{n(n+1)}>n, then n+\tfrac{1}{2}-\sqrt{n(n+1)}=\tfrac{n+(n+1)}{2}-\sqrt{n(n+1)} can be re-written as n+\tfrac{1}{2}-n>\tfrac{n+(n+1)}{2}-\sqrt{n(n+1)}. (This is a more constrictive inequality and it is true, it’s just a bit tricky to visualise; make sure this makes sense because it took me a while to figure it out.)

Combining the above two arguments, we get [0\leq\tfrac{n+(n+1)}{2}-\sqrt{n(n+1)}<\frac{1}{2}.]

It took a while, but we have something meaningful here! Now we know that (\sqrt{n}+\sqrt{n+1})^2=4n+2-2\left(\frac{n+(n+1)}{2}+\sqrt{n(n+1)}\right) can be re-written as [\left(\sqrt{n}+\sqrt{n+1}\right)^2=4n+2-\text{[some value in between 0 and 1, including 0]}.]

So, [4n+1<\left(\sqrt{n}+\sqrt{n+1}\right)^2\leq 4n+2.]

Recall to above when we mentioned that the problem is essentially proving that there is no perfect square between \left(\sqrt{n}+\sqrt{n+1}\right)^2 and \sqrt{4n+2}^2=4n+2. Using above, this boils down to proving that no perfect square fits between 4n+2 and 4n+1. (This is the ‘hardest’ or most (or least? idek, lol) restrictive case of the inequality.)

Obviously, there aren’t any perfect square between 4n+1 and 4n+2 because they are consecutive integers. So, there are no numbers between \sqrt{n}+\sqrt{n+1} and \sqrt{4n+2}. Thus, \lfloor\sqrt{n}+\sqrt{n+1}\rfloor=\lfloor\sqrt{4n+2}\rfloor and we are done. \blacksquare

Nice problem, btw. (Also, I guess align doesn’t quite work, . If someone knows how to get it to work, that would be great, thanks! In the meantime, look here for pretty \LaTeX.)

A nice proof, with a different approach than mine. You have fewer cases to consider, which is good.

That’s not quite good enough - if the larger side were an integer, the two floors would be unequal. However as all squares are of the form 4n or 4n+1, the right hand side can’t be an integer. It doesn’t seem likely that \sqrt{n}+\sqrt{n+1} can be an integer for n\in\mathbb{N}, n>1, but it doesn’t seem easy to prove. It’s easier to prove that the left hand side is the smaller*; it doesn’t matter if the smaller side is an integer.

Added: I figured out how to show that \sqrt{n}+\sqrt{n+1} is not an integer. The square of an integer will be an integer. Square it, and after rearranging you get: 2n+1+\sqrt{4n^2+n}. The first two terms are integers, so the last one must be as well. But 4n^2+4n+1=(2n+1)^2, so the 4n^2+4n is one less than a perfect square. The only perfect squares separated by one are zero and one, but this would imply n=0, but 0\notin\mathbb{N}.

Later: I saw the doc you linked. There you stated that it was necessary to show that there was not an integer between the values, or that neither was an integer. You actually need to prove BOTH that there are no integers between, and that the larger is not an integer.

* This is actually lemma 1 in my proof below.

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My Proof:
For convenience and clarity, let’s define some real functions for the arguments of the floor function. We’ll limit the domains (inputs) to positive values to simplify later steps.
\begin{align*} f(x)&:= \sqrt{x}+\sqrt{x+1}, &x>0 \\ g(x)&:=\sqrt{4x+2}, &x>0 \end{align*}

As x>0, it is easy to see that: f(x) > 0; f'(x)>0; g(x)>0; g'(x)>0. That is, both f(x) and g(x) are positive and increasing with increasing x.

Lemma 1: f(x)<g(x) for all x>0.
As both functions are positive, we can compare them by comparing their squares.
\begin{align*} f^2(x)=(\sqrt{x}+\sqrt{x+1})^2 = x + 2\sqrt{x^2+x}+x+1=&2x+1+\sqrt{4x^2+4x} \\ g^2(x)=4x+2=2x+1+(2x+1)=&2x+1+\sqrt{4x^2+4x+1} \end{align*}

Looking at these it is clear that f^2(x)<g^2(x), therefore f(x)<g(x).

Inspecting tabulated values for \lfloor{f(n)}\rfloor and \lfloor{g(n)}\rfloor (thanks @gerthworm!), it appears that these are indeed equal, and increase for numbers which can be expressed as m^2 or m^2+m, m\in{\mathbb{N}}.
Based on this, we define m:=\lfloor\sqrt{n}\rfloor.

Lemma 2: m^2\leq n\leq m^2+m.
As m=\lfloor\sqrt{n}\rfloor\leq\sqrt{n}, m^2\leq{n}. Also, n< m^2+2m+1 because

n\geq m^2+2m+1\implies\sqrt{n}\geq m+1\implies \sqrt{n}\geq \lfloor{\sqrt{n}}\rfloor+1

which clearly false. But this implies n\leq m^2+m, proving both parts of the lemma.

Conjecture, from tabulation:
\begin{align*} m^2\leq &n \leq m^2+m-1:&&\lfloor{f(n)}\rfloor=\lfloor{g(n)}\rfloor=2m \\ m^2+m\leq &n \leq m^2+2m:&&\lfloor{f(n)}\rfloor=\lfloor{g(n)}\rfloor=2m+1 \end{align*}
From Lemma 2, these two cases combine to include all n\in\mathbb{N}.

As we have already shown that f(x) and g(x) are increasing with increasing x, and that f(x)<g(x), we need to prove two things for each half of the conjecture, in particular:
\begin{align*} &1: &2m&\leq f(m^2) \\ &2: &g(m^2+m-1)&<2m+1 \\ &3: &2m+1&\leq f(m^2+m) \\ &4: &g(m^2+2m)&<2m+2 \end{align*}

Actually, let’s go a little better and prove them all as strict inequalities. In case 3, we use that both f(x) and 2m+1 are positive to take the square root of the square.
\begin{align*} 1: f(m^2)&=\sqrt{m^2}+\sqrt{m^2+1} \\ &>\sqrt{m^2}+\sqrt{m^2} \\ &>2m \end{align*}
\begin{align*} 2: g(m^2+m-1)&=\sqrt{4m^2+4m-4+2} \\ &<\sqrt{4m^2+4m+1} \\ &<2m+1 \end{align*}
\begin{align*} 3: f(m^2+m)&=\sqrt{m^2+m}+\sqrt{m^2+m+1} \\ &=\sqrt{(\sqrt{m^2+m}+\sqrt{m^2+m+1})^2} \\ &=\sqrt{m^2+m +2\sqrt{m^2+m}\sqrt{m^2+m+1}+m^2+m+1}\\ &=\sqrt{2m^2+2m+1 +2\sqrt{m^2+m}\sqrt{m^2+m+1}}\\ &>\sqrt{2m^2+2m+ 1+2\sqrt{m^2+m}\sqrt{m^2+m}}\\ &>\sqrt{2m^2+2m+ 1+2(m^2+m)}\\ &>\sqrt{4m^2+4m+ 1}\\ &>\sqrt{(2m+1)^2}\\ &>2m+1 \end{align*}
\begin{align*} 4: g(m^2+2m)&=\sqrt{4m^2+8m+2}\\ &<\sqrt{4m^2+8m+4}\\ &<2m+2 \end{align*}

As we have proven the conjecture, we have also proven the goal, because the two parts combine to prove \lfloor{f(n)}\rfloor=\lfloor{g(n)}\rfloor for every natural number.

Make sure you wrapped the align block in a math environment via dollar signs like so.

\begin{align*} 2^2 + 1 &= 5 \\ 3^2 &= 9 \end{align*}


\begin{align*} 2^2 +1 &= 5 \\ 3^2 &= 9 \end{align*}

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Did you mean?
x-1<⌊x⌋≤x

Actually, I meant x\ge\lfloor{x}\rfloor\lt x+1, but what you wrote is also correct. I have fixed it above.

Now it’s really confusing with signs facing both directions…
if ⌊x⌋\le x then ⌊x⌋<x+1 follows naturally, therefore you just wrote ⌊x⌋\le x (or I didn’t understand you correctly).

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That’s actually what I meant…to “squeeze” the value, but yes, yours is less confusing… I’ll go to that.

Yes! @Ether established several “Quiz” lines, and the idea was always for people to post their solutions, or even their conjectures. Sometimes posting was limited to students, but usually was open to all the denizens of CD. Search “@ether Quiz” (no quotes) to find the old problems. I’m trying to follow in his footsteps and hone skills across FRC. My personal favorites were
Math Quiz 8
Math Quiz 9
But all of them are worth the trip. Unfortunately, a number of them were severely damaged in the port to Discourse.

It doesn’t work with LaTeX, but for most posts, if you want to hide text, as you’re entering or editing, select the text, then click on the “Gear” above the post and release over “Hide Details”. Change the word “Summary” in what comes up to something meaningful. Here I changed it to “Useless text to be hidden”. If you want to see what it looks like internally, select the rest of this post and select “Quote”.

Useless text to be hidden

Let’s Hide This Text

While I don’t want to make a ton more work for the mods, would it be useful to dump these in their own category/channel under Other? I’d be happy to toss out a question/quiz every now and then… see if the group here can collectively continue @Ether’s legacy and keep the fun problems coming!

Dr. Michael Penn posted a video with another (more elegant) solution to this problem yesterday.

Synopsis, with a bit of license to make it even slicker IMO:

Note that as 4n+2 and 4n+3 (and non-integers) cannot be perfect squares, the goal statement is true if:

\sqrt{4n+1} \le \sqrt{n} + \sqrt{n+1} < \sqrt{4n+4}

As all terms are clearly positive, he then squares all sides, subtracts 2n+1* from each side, and gathers terms, yielding:

2n \le 2\sqrt{n^2 + n} < 2n+3

Now, let’s square all three* again, yielding

4n^2\le4n^2+4n<4n^2+12n+9

and in this form, both inequalities are pretty obvious as all natural numbers are greater than zero.   * Here’s where I take license. He subtracted 4n+1, and later had to add 2n back for the right hand inequality, and only squared the central and right terms.

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