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Let $A_n$ be the sum of the first $n$ prime numbers. Prove that there is a perfect square between $A_n$ and $A_{n+1}$.

This is how I prove the conjucture:

Using Proof by contradiction, suppose there is no perfect square between A_n and A_(n+1).

It implies that k^2 < A_n and (〖k+1)〗^2>A_(n+1) for any natural number k.

Let A_n=p_1+p_2+p_3+⋯+p_n A_(n+1)=p_1+p_2+p_3+⋯+p_(n+1)

The difference between (〖k+1)〗^2and k^2 is 2k+1 and the difference between A_n and A_(n+1)is p_(n+1), thus p_(n+1)<2k+1.

Since k^2= 1+3+5+...+(2k-1), thus p_n ≥ 2k-1 and p_(n+1) ≥ 2k+1. It then contradicts p_(n+1)<2k+1.

Therefore there is a perfect square between A_n and A_(n+1)where A_n is the sum of the first n prime numbers.

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  • $\begingroup$ including them? $\endgroup$ – evil999man Apr 21 '14 at 16:47
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    $\begingroup$ @user144691, just FYI you should generally wait at least a day before accepting an answer. $\endgroup$ – 6005 Apr 21 '14 at 17:05
  • $\begingroup$ What is the source of the problem? What other material is discussed in the book before this exercise? $\endgroup$ – Will Jagy Apr 21 '14 at 18:04
  • $\begingroup$ Judging from this other post, this OP is asking questions without knowing whether they are true or false, although worded as though a proof is available and easy: math.stackexchange.com/questions/763147/… Voting to close $\endgroup$ – Will Jagy Apr 21 '14 at 18:10
  • $\begingroup$ What is your argument for $p_n \geqslant 2k-1$? $\endgroup$ – Daniel Fischer Jul 20 '14 at 10:57
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This statement appears to be true, and could perhaps be proved with fairly basic analytic estimates, but it is by no means an elementary question. Here is data up to the point where the sum of consecutive primes exceeds 10,000. Note that, when the sum is exactly 100, the square 81 could have been chosen instead. Posted with CW status as I voted to close the question.


  April 21  
                             1
    2         2
                             4
    3         5
                             9
    5        10
                            16
    7        17
                            25
   11        28
                            36
   13        41
                            49
   17        58
                            64
   19        77
                           100
   23       100  =-=-=-=-=
                           121
   29       129
                           144
   31       160
                           196
   37       197
                           225
   41       238
                           256
   43       281
                           324
   47       328
                           361
   53       381
                           400
   59       440
                           484
   61       501
                           529
   67       568
                           625
   71       639
                           676
   73       712
                           784
   79       791
                           841
   83       874
                           961
   89       963
                          1024
   97      1060
                          1156
  101      1161
                          1225
  103      1264
                          1369
  107      1371
                          1444
  109      1480
                          1521
  113      1593
                          1681
  127      1720
                          1849
  131      1851
                          1936
  137      1988
                          2116
  139      2127
                          2209
  149      2276
                          2401
  151      2427
                          2500
  157      2584
                          2704
  163      2747
                          2809
  167      2914
                          3025
  173      3087
                          3249
  179      3266
                          3364
  181      3447
                          3600
  191      3638
                          3721
  193      3831
                          3969
  197      4028
                          4225
  199      4227
                          4356
  211      4438
                          4624
  223      4661
                          4761
  227      4888
                          5041
  229      5117
                          5329
  233      5350
                          5476
  239      5589
                          5776
  241      5830
                          5929
  251      6081
                          6241
  257      6338
                          6561
  263      6601
                          6724
  269      6870
                          7056
  271      7141
                          7396
  277      7418
                          7569
  281      7699
                          7921
  283      7982
                          8100
  293      8275
                          8464
  307      8582
                          8836
  311      8893
                          9025
  313      9206
                          9409
  317      9523
                          9801
  331      9854
                         10000
  337     10191

The thing would appear to follow from effective bounds on the size of the $n$-th prime in Rosser and Schoenfeld (1962) available online. The resulting inequality confirms the statement for a specific lower bound $n \geq N,$ and a computer run confirms it below $n.$ I expect i will figure out $N$ tomorrow, it is not immediate...

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