Problem of the Month
(February 2014)

A matrix whose rows and columns are all different primes with non-decreasing digits is called a increasing prime grid. Similarly, if the primes have non-increasing digits, we call it a decreasing prime grid. For small m and n, can you find all the m×n increasing/decreasing prime grids? For a fixed m, what is the largest n so that an increasing/decreasing prime m×n grid exists? What is the largest increasing/decreasing prime grid that you can find?


ANSWERS

Solutions were received by Joe DeVincentis, Bryce Herdt, and Hakan Summakoğlu.

Here are the known increasing prime grids of different sizes:

1×2
23   37

1×3
257

1×4
2357

2×2
11   11
37   79

2×3
113   127   127   137   137   157   167   167
137   139   199   179   379   199   179   379

2×4
1117   1367   2347   2467
1399   3779   3779   3779

2×5
11117   11257   11257   11257   12457   13457
13799   13399   13999   79999   13799   37799

2×6
111127   113467
137999   377779

3×4
1123   1223   2333   2333   2333   2333
1277   1277   2347   2467   3347   3467
3779   3779   3779   3779   3779   3779

3×5
11113   11113   11113   11113   23333   23333
12347   12377   12577   13577   23447   33457
37799   37799   37799   37799   37799   37799

3×6
112223
122267
377999
(JD)

4×5
11113   11113   11113   11113   11113   11113
14557   12227   12227   12227   12457   11257
24677   13457   13477   13577   13577   12277
37799   77999   77999   77999   77999   79999

11113   11113   11113   11113   11113   11113
11257   12227   11257   12227   12457   12457
12457   12457   12577   12577   12577   14557
79999   79999   79999   79999   79999   79999
(JD)

4×6
112223   112223   112223   112223   112223   112223
123377   122347   122477   222337   222347   222367
223577   233477   233477   233477   233477   233477
333779   377779   377779   377779   377779   377779

112223   112223   112223   112223   112223   112223
122347   122477   222347   222367   123457   223577
233777   233777   233777   233777   234457   235577
377779   377779   377779   377779   377779   377779

112223   112223   112223   112223   112223   112223
122347   222247   222347   123457   122347   222347
233357   233357   233357   235577   233357   233357
377999   377999   377999   377999   379999   379999
(JD)

5×5
864 solutions
(JD)

4×7
1112333
1223357
2235557
3377999
(JD)

5×6
13349 solutions
(JD)

4×8
11122333
12223457
22333457
33799999
(JD)

5×7
10489 solutions
(JD)

6×6
5377 solutions
(JD)

5×8
135 solutions
(JD)

6×7
136374 solutions
(JD)

5×9
81253 solutions
(HS)

7×7
2304 solutions
(JD)

7×8
13891 solutions
(JD)

Here are the number of increasing prime grids of various sizes:

×12345678
10
222
3180
41460
506612864
602118133495377
70001104891363742304
800011350138910
9000081253


Here are the decreasing prime grids of different sizes, up to symmetry:

1×2
53   73

2×2
41   71
31   31

2×3
431   443   541   541   743   743
311   311   311   331   311   331

751   761   773   853   863   877
331   311   311   331   311   331

2×4
7541   8543   8741   8753   9743   9851   9871
3331   3331   3331   3331   7331   7331   7331

3×3
643   743   853
541   733   643
331   311   311

2×5
75431   87443   87541   87541   98543
33311   33311   33311   33331   73331

2×6
876431   974431   987541   987631
311111   733111   733331   733111

3×4
7643   7643   7643   8443   8443   8443   8443   8663   8663
4441   7433   7541   5431   7433   8431   8431   7541   8543
3331   3331   3331   3331   7331   3331   7331   7331   3331

8663   8863   8863   8863   8863   8863   8863   8863   8863
8543   6551   6553   7541   7643   8543   8543   8641   8641
7331   3331   3331   7331   7331   3331   7331   3331   7331

8887   9643   9643   9643   9643   9643   9643   9643   9643
8863   5431   5441   5443   5531   7433   7541   8431   8443
7333   3331   3331   3331   3331   7331   7331   3331   3331

9643   9643   9643   9643   9743   9743   9743   9743   9743
8543   9431   9433   9533   5333   5431   5441   5443   7333
3331   7331   7331   7331   3331   3331   3331   3331   7331

9743   9743   9743   9743   9743   9743   9743   9743   9743
7433   7741   8431   8443   8731   8741   9431   9433   9733
7331   7331   3331   3331   3331   3331   7331   7331   7331

9883   9883   9883   9973   9973   9973   9973   9973       
8861   8863   9851   7541   8543   9533   9833   9871       
3331   3331   7331   7331   3331   7331   7331   7331       

2×7
9875443
7333311

3×5
64433   84443   84443   87433   87433   87433   87433   87433
54331   54331   64333   54331   63331   84431   54331   54421
33311   33311   33311   33311   33311   33311   33311   33311

87443   87443   87443   87443   87443   87443   87443   87443
63331   64333   64433   64433   77431   84421   84431   84431
33311   33311   33311   33331   73331   33311   33311   33331

87443   87443   87443   87443   87443   87643   87643   87643
84431   87421   87433   87433   87433   54421   54443   64433
73331   33311   33311   33331   73331   33311   33331   33311

87643   87643   87643   87643   87643   87643   87643   87643
64433   74441   77431   77543   84421   84431   84431   84431
33331   73331   73331   73331   33311   33311   33331   73331

87643   87643   87643   87643   87643   87643   87643   87643
84443   84443   87421   87433   87433   87433   87443   87443
33331   73331   33311   33311   33331   73331   33331   73331

87643   87643   87743   87743   87743   87743   87743   87743
87541   87541   54331   54331   64333   64333   77431   87421
33331   73331   33311   33331   33311   33331   73331   33311

87743   87743   87743   87743   87743   88643   88643   88643
87433   87433   87433   87443   87443   65521   65543   75431
33311   33331   73331   33331   73331   33311   33331   73331

88643   88643   88643   88643   88643   88643   88643   88643
75533   75541   76441   76541   76543   85531   85531   85531
73331   73331   73331   73331   73331   33311   33331   73331

88643   88643   88643   88643   88643   88643   88643   88643
86441   86441   86531   86531   86531   86533   86533   86533
33331   73331   33311   33331   73331   33311   33331   73331

88663   88663   88663   88663   88663   88663   88663   88663
65543   65543   75541   76541   76543   85531   86441   86531
33311   33331   73331   73331   73331   33311   33311   33311

88663   88843   88843   88843   88843   88843   88843   88843
86533   76541   76543   86531   86531   86531   86533   86533
33311   73331   73331   33311   33331   73331   33311   33331

88843   88843   88873   88873   88873   88873   88873   88873
86533   88643   76541   76543   86531   86531   86533   86533
73331   73331   73331   73331   33331   73331   33331   73331
  
88873   88883   94433   96443   96443   96443   96443   96443
88643   88651   54331   54331   54421   55331   55333   75431
73331   73331   33311   33311   33311   33311   33311   73331

96443   96443   96443   96443   96443   96443   96643   96643
84421   84431   84431   85331   85333   94433   55441   75431
33311   33311   33331   33311   33311   73331   33331   73331

96643   96643   98443   98443   98443   98443   98443   98443
95441   95443   55331   55333   75431   85331   85333   88321
73331   73331   33311   33311   73331   33311   33311   33311

98443   98663   98663   98663   98663   98663   98663   98663
96431   55441   55541   55541   75541   76541   76543   85531
73331   33311   33311   33331   73331   73331   73331   33311

98663   98663   98663   98663   98773   98773   98773   98773
86441   86531   86533   98543   75431   88741   96431   98731
33311   33311   33311   73331   73331   33331   73331   73331
   
98873   98873   98873   98873   98873   98873   98873   98873
76541   76543   86531   86533   88643   88651   88661   88663
73331   73331   33331   33331   33331   33311   33311   33311

98873   98873   98873   98873   98873   99643   99643   99643
98533   98543   98641   98731   98773   75431   75533   75541
73331   73331   73331   77731   77731   73331   73331   73331

99643   99643   99643   99643   99643   99643   99643   99643
85531   85531   95441   95443   95531   98443   98533   98543
33311   33331   73331   73331   73331   73331   73331   73331

99733   99833                                                
85331   85531                                                
33311   33311                                                

4×4
9887
9643
7643
3331
(JD)

3×6
194 solutions
(JD)

4×5
99877   98773   97777   98773   97777   98773
88663   87433   87433   87433   87433   87433
88643   85333   85333   85333   85333   85333
73331   73331   73331   33331   33331   33311

99877   98887   98873   98663   88643        
87553   98773   98773   97553   87643        
84443   76543   76543   75553   65543        
33311   33311   33311   33311   33311        
(JD)

3×7
168 solutions
(JD)

4×6
49 solutions
(JD)

5×5
98663   98773   88853   88663   98663
88643   88663   88843   87443   96643
86533   86533   88643   87433   65543
86531   84431   87641   54331   43331
73331   73331   33331   33331   33331

98663   96643   97777   88663   88643
88663   86533   87553   87433   87433
88643   66533   65543   85333   85333
75533   64433   64333   55333   55333
33311   33311   33311   33311   33311
(JD)

3×8
98877533   98866643   98866433   99877643   99877633
98765431   98764321   98743331   97765331   97765331
77711111   77711111   77711111   77711111   77711111

99877643   99876443   99876443   99877643   98766433
97765321   97753321   99866321   99865321   98664331
77711111   77711111   71111111   71111111   71111111

98776543   98766433   99766553   99965443   97766443
98654221   98643331   77664421   77444321   76543321
71111111   71111111   71111111   71111111   71111111
(JD)

3×9
56 solutions
(JD) (HS)

4×7
24 solutions
(JD)

5×6
899 solutions
(JD)

4×8
99988853   98887753   98877553
88887443   88777543   88775443
87533333   86644433   86644433
71111111   71111111   71111111
(JD)

5×7
1228 solutions
(JD)

6×6
996 solutions
(JD)

5×8
335 solutions
(JD)

6×7
11352 solutions
(JD)

5×9
1254 solutions
(JD) (HS)

6×8
5408 solutions
(JD)

7×7
6279 solutions
(JD)

6×9
20066 solutions
(JD) (HS)

7×8
1669 solutions
(JD)

7×9
101901 solutions
(JD)

Here are the number of decreasing prime grids of various sizes:

×12345678
10
222
30123
407441
5051621110
60419449899996
701168241228113526279
800153335540816690
900560125420066101901


Both Joe DeVincentis and Bryce Herdt generalized to grids that were decreasing across and increasing down.

1×2
53   73

2×1
2   3
3   7

1×3
2
5
7

1×4
2
3
5
7

2×2
31   41   61   11   31   41
71   71   71   73   73   73

61   11   53   71   73   83
73   97   97   97   97   97
(BH)

2×3
211   421   311   431   631   641   211
331   733   773   773   773   773   971

541   761   811   541   743   761   863
971   971   971   977   977   977   977

211   521   751   811   821   211   521
991   991   991   991   991   997   997

751   811   821   853                  
997   997   997   997                  
(JD)

2×4
4211   6211   7321   7621   2111   7211   8111   8221
7331   7331   9733   9733   9931   9931   9931   9931

8521   2111   7211   7541   8111   8731   8741   8761
9931   9973   9973   9973   9973   9973   9973   9973
(JD)

4×2
31   41
41   61
61   71
97   97
(JD)

3×3
67 solutions
(JD)

2×5
54311   74311   86311   87421   75211   87211
97771   97771   97771   99733   99971   99971

87511   87541   75211   87211   87511        
99971   99971   99991   99991   99991        
(JD)

5×2
11   11   11   31   41   31   41
31   31   31   41   43   43   43
41   41   61   61   53   53   73
71   61   71   71   73   83   83
73   97   97   97   97   97   97
(JD)

2×6
852211   752111
999331   999931
(JD)

6×2
11   11   41   31
31   31   43   61
41   41   53   71
43   73   73   73
83   83   83   83
97   97   97   97
(JD)

3×4
50 solutions
(JD)

4×3
58 solutions
(JD)

2×7
8643211
9777331
(JD)

7×2
11
31
41
43
53
83
97
(JD)

3×5
38 solutions
(JD)

5×3
265 solutions
(JD)

3×6
33 solutions
(JD)

6×3
284 solutions
(JD)

4×5
140 solutions
(JD)

5×4
370 solutions
(JD)

3×7
24 solutions
(JD)

7×3
205 solutions
(JD)

3×8
55421111   54432211   54331111   54331111   54333311
96433321   97443221   97744321   97753321   97754431
99999773   99999773   99999773   99999773   99999773

43321111   43333331   53333311   43321111   54432211
98533321   98754431   98754431   98764321   99443221
99999773   99999773   99999773   99999773   99999773

54331111   54322211   54331111                      
99544321   99732221   99743321                      
99999773   99999773   99999773                      
(JD)

8×3
204 solutions
(JD)

5×5
417 solutions
(JD)

3×9
544322111
997775531
999977773
(HS)

9×3
112 solutions
(HS)

4×7
618 solutions
(JD)

7×4
295 solutions
(JD)

5×6
3997 solutions
(JD)

6×5
232 solutions
(JD)

5×7
15914 solutions
(JD)

7×5
185 solutions
(JD)

9×4
262 solutions
(HS)

5×8
12202 solutions
(JD)

8×5
418 solutions
(JD)

6×7
35270 solutions
(JD)

7×6
38435 solutions
(JD)

5×9
8122 solutions
(HS)

9×5
71 solutions
(HS)

7×7
268113 solutions
(JD)

7×8
247700 solutions
(JD)

8×7
129164 solutions
(JD)

7×9
8122 solutions
(HS)

Here are the number of increasing/decreasing prime grids of various sizes:

inc\dec123456789
1020000000
22122516112100
3106750383324131
412580140061800
507265370417399715914122028122
604284023203427000
70120529518538435268113247700564858
800204041801291640
90011226271


If you can extend any of these results, please e-mail me. Click here to go back to Math Magic. Last updated 2/23/14.