\( \def\pm{{ ‰}} \def\pmf{{ ‰ \phantom8}} \def\pmm{{ ‰ \! ‰}} \def\pmmf{{ ‰ \! ‰ \phantom\%}} \DeclareMathOperator{\antilog}{antilog} \DeclareMathOperator{\arcsec}{arcsec} \DeclareMathOperator{\csch}{csch} \)

Integermania!

First Four Primes

The first four prime numbers are 2, 3, 5, and 7. Create each of the positive integers using one copy of each number, and any standard operations. All four numbers must be used, but no others. Your solutions will be assigned an exquisiteness level.

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      4093 (3.0)
$2^{7+5} - 3$
Jackson Jennings, 3/15
Shawnee, KS		           4099 (3.0)
$2^{7+5} + 3$
Jackson Jennings, 3/15
Shawnee, KS		  
                    4320 (2.2)
$\dfrac{72}{5\%} \times 3$
Dave Jones, 2/07
Coventry, England
          4375 (3.2)
$5^{3!-2} \times 7$
Phaedra Dowell, 4/05
Roeland Park, KS		          
                    4480 (2.2)
$\dfrac{32}{5\%} \times 7$
Dave Jones, 2/07
Coventry, England
                    4500 (2.2)
$\dfrac{27}{3\%} \times 5$
Dave Jones, 2/07
Coventry, England
                    4740 (2.2)
$\dfrac{237}{5\%}$
Dave Jones, 2/07
Coventry, England
              4807 (5.0)
$\dfrac{ {}_{23} C_5}{7}$
Lucas Rodriguez, 12/04
Lenexa, KS		      
    4912 (3.6)
$7! - 5! - 3! - 2$
Brooke Jones, 4/04
Olathe, KS		                
                    5000 (2.2)
$\dfrac{75}{3\%} \times 2$
Dave Jones, 2/07
Coventry, England
                    5040 (3.2)
$7! + 5 - 3 - 2$
Steve Wilson, 1/09
Raytown, MO
                    5070 (3.2)
$7! + 5 \times 3 \times 2$
Brooke Jones, 4/04
Olathe, KS
    5152 (3.6)
$7! + 5! - 3! - 2$
Brooke Jones, 4/04
Olathe, KS		                
        5164 (3.6)
$7! + 5! + 3! - 2$
Wei Zhang, 7/12
Lawrence, KS		       5168 (3.6)
$7! + 5! + 3! + 2$
Andy Powers, 5/02
Basehor, KS		    
    5172 (3.6)
$7! + 5! + 3! \times 2$
Ron Palcic, 8/04
Olathe, KS		                
              5237 (2.0)
$5237$
Dave Jones, 8/06
Coventry, England		      
                    5250 (2.2)
$\dfrac{7}{2\%} \times 5 \times 3$
Dave Jones, 2/07
Coventry, England
      5273 (2.0)
$5273$
Dave Jones, 8/06
Coventry, England		              
          5285 (3.2)
$7! + 3^5 + 2$
Phaedra Dowell, 1/05
Roeland Park, KS		          
              5327 (2.0)
$5327$
Dave Jones, 8/06
Coventry, England		      
        5334 (3.0)
$73^2 + 5$
Brittney King, 9/07
Olathe, KS		            
    5372 (2.0)
$5372$
Dave Jones, 8/06
Coventry, England		                
                    5460 (2.2)
$\dfrac{273}{5\%}$
Dave Jones, 2/07
Coventry, England
      5603 (3.0)
$\dfrac{7^5 + 2}{3}$
Yuncan Yang, 7/12
Lawrence, KS		              
                5628 (3.0)
$75^2 + 3$
John Kite, 8/02
Lenexa, KS		    
      5723 (2.0)
$5723$
Dave Jones, 8/06
Coventry, England		              
    5732 (2.0)
$5732$
Dave Jones, 8/06
Coventry, England		                
          6125 (3.0)
$7^2 \times 5^3$
Bridget Osei-Bonsu, 11/04
Lenexa, KS		          
                    6540 (2.2)
$\dfrac{327}{5\%}$
Dave Jones, 2/07
Coventry, England
  6561 (3.0)
$3^5 \times 27$
Adam McAdoo, 5/05
Lenexa, KS		                  
        6784 (3.0)
$2^7 \times 53$
Kimberly Grosdidier, 12/05
Eudora, KS		            
                    7000 (3.0)
$(2 \times 5)^3 \times 7$
David Caton, 9/04
Fort Scott, KS
          7235 (2.0)
$7235$
Dave Jones, 8/06
Coventry, England		          
      7253 (2.0)
$7253$
Dave Jones, 8/06
Coventry, England		              
          7325 (2.0)
$7325$
Dave Jones, 8/06
Coventry, England		          
    7352 (2.0)
$7352$
Dave Jones, 8/06
Coventry, England		                
                    7440 (2.2)
$\dfrac{372}{5\%}$
Dave Jones, 2/07
Coventry, England
      7523 (2.0)
$7523$
Dave Jones, 8/06
Coventry, England		              
    7532 (2.0)
$7532$
Dave Jones, 8/06
Coventry, England		                
                    7600 (2.2)
$\dfrac{532}{7\%}$
Dave Jones, 2/07
Coventry, England
                    8550 (2.2)
$\dfrac{57}{2\%} \times 3$
Dave Jones, 2/07
Coventry, England
          8575 (3.0)
$35^2 \times 7$
Ashley Brewer, 2/02
Olathe, KS		          
                    9000 (3.0)
$72 \times 5^3$
Apollo Okwuone, 12/04
Olathe, KS
                    9250 (2.2)
$\dfrac{37}{2\%} \times 5$
Dave Jones, 2/07
Coventry, England
              9747 (3.0)
$57^2 \times 3$
Brittney King, 12/07
Olathe, KS		      
          9955 (3.2)
$7! \times 2 - 5^3$
Brooke Jones, 5/04
Olathe, KS		          
                    10080 (3.4)
$7 \times 5! \times 3! \times 2$
Wei Zhang, 7/12
Lawrence, KS
                    10800 (3.6)
$7! \times 2 + 3! \times 5!$
Brooke Jones, 5/04
Olathe, KS
            10976 (3.0)
$2^5 \times 7^3$
Brittney King, 10/07
Olathe, KS		        
          11025 (3.0)
$(7 \times 5 \times 3)^2$
Kashmira Sayani, 3/17
Overland Park, KS		          
                    11250 (2.2)
$\dfrac{75}{2\%} \times 3$
Dave Jones, 2/07
Coventry, England
                    12000 (2.2)
$\dfrac{72}{3\%} \times 5$
Dave Jones, 2/07
Coventry, England
          12005 (3.2)
$7^{3!-2} \times 5$
Phaedra Dowell, 5/05
Roeland Park, KS		          
                    12250 (2.2)
$\dfrac{35}{2\%} \times 7$
Dave Jones, 2/07
Coventry, England
                    13720 (3.0)
$(7 \times 2)^3 \times 5$
David Caton, 10/04
Shawnee, KS
                    14460 (2.2)
$\dfrac{723}{5\%}$
Dave Jones, 2/07
Coventry, England
                    14640 (2.2)
$\dfrac{732}{5\%}$
Dave Jones, 2/07
Coventry, England
                  15309 (3.0)
$7 \times 3^{5+2}$
Alyssa Trembly, 4/15
Spring Hill, KS		  
                15478 (3.4)
$(7! + 5!) \times 3 - 2$
Long Zhang, 7/12
Lawrence, KS		    
              15497 (3.2)
$5^{3!} - 2^7$
Phaedra Dowell, 3/05
Roeland Park, KS		      
                15618 (3.0)
$25^3 - 7$
Kimberly Grosdidier, 12/05
Eudora, KS		    
    15632 (3.0)
$5^{3 \times 2} + 7$
Lucas Rodriguez, 11/04
Lenexa, KS		                
      15753 (3.2)
$5^{3!} + 2^7$
Phaedra Dowell, 3/05
Roeland Park, KS		              
                    16000 (3.0)
$2^7 \times 5^3$
Cate Henderson, 5/05
Merriam, KS
        16384 (3.0)
$\left( \dfrac{5+3}{2} \right)^7$
Allan Hackett, 2/04
Overland Park, KS		            
                16798 (3.0)
$7^5 - 3^2$
Steve Wilson, 2/07
Raytown, MO		 16799 (3.0)
$7^5 - 2^3$
Steve Wilson, 2/07
Raytown, MO		  
  16801 (3.0)
$7^5 - 3 \times 2$
Steve Wilson, 2/07
Raytown, MO		 16802 (3.0)
$7^5 - 3 - 2$
Steve Wilson, 2/07
Raytown, MO		 16803 (3.2)
$7^5 - 3! + 2$
Steve Wilson, 2/07
Raytown, MO		 16804 (3.2)
$7^5 - \dfrac{3!}{2}$
Steve Wilson, 2/07
Raytown, MO		   16806 (3.0)
$7^5 - 3 + 2$
Steve Wilson, 2/07
Raytown, MO		 16807 (3.0)
$7^5 \times (3 - 2)$
Steve Wilson, 2/07
Raytown, MO		 16808 (3.0)
$7^5 + 3 - 2$
Bridget Osei-Bonsu, 12/04
Lenexa, KS		   16810 (3.2)
$7^5 + \dfrac{3!}{2}$
Steve Wilson, 2/07
Raytown, MO
  16811 (3.2)
$7^5 + 3! - 2$
Phaedra Dowell, 1/05
Roeland Park, KS		 16812 (3.0)
$7^5 + 3 + 2$
Steve Wilson, 2/07
Raytown, MO		 16813 (3.0)
$7^5 + 3 \times 2$
Steve Wilson, 2/07
Raytown, MO		   16815 (3.0)
$7^5 + 2^3$
Chad Schipper, 4/03
Prairie Village, KS		 16816 (3.0)
$7^5 + 3^2$
Steve Wilson, 2/07
Raytown, MO		        
  16871 (3.2)
$7^5 + 2^{3!}$
Phaedra Dowell, 3/05
Roeland Park, KS		       16875 (3.0)
$75^2 \times 3$
Katie Cooper, 10/04
Stilwell, KS		          
                    17850 (2.2)
$\dfrac{357}{2\%}$
Dave Jones, 2/07
Coventry, England
                    18250 (2.2)
$\dfrac{73}{2\%} \times 5$
Dave Jones, 2/07
Coventry, England
                    18550 (2.2)
$\dfrac{53}{2\%} \times 7$
Dave Jones, 2/07
Coventry, England
                    18750 (2.2)
$\dfrac{375}{2\%}$
Dave Jones, 2/07
Coventry, England
                19678 (3.0)
$27^3 - 5$
Jared Northrup, 4/03
Lawrence, KS		    
                19688 (3.0)
$27^3 + 5$
Rabeh Ghadiri, 9/08
Overland Park, KS		    
                    21870 (3.0)
$2 \times 5 \times 3^7$
Brittney King, 10/07
Olathe, KS
          21875 (3.0)
$7 \times 5^{3+2}$
Alyssa Trembly, 4/15
Spring Hill, KS		          
          26645 (3.0)
$73^2 \times 5$
Brittney King, 12/07
Olathe, KS		          
                    26850 (2.2)
$\dfrac{537}{2\%}$
Dave Jones, 2/07
Coventry, England
                    28650 (2.2)
$\dfrac{573}{2\%}$
Dave Jones, 2/07
Coventry, England
                28768 (5.0)
$\dfrac{ {}_{32} C_5}{7}$
Lucas Rodriguez, 12/04
Lenexa, KS		    
  29241 (3.0)
$(57 \times 3)^2$
Kashmira Sayani, 1/17
Overland Park, KS		                  
                    30960 (3.4)
$(7! + 5!) \times 3 \times 2$
Wei Zhang, 7/12
Lawrence, KS
        31104 (3.0)
$2^7 \times 3^5$
Bridget Osei-Bonsu, 11/04
Lenexa, KS		            
              33617 (3.0)
$2 \times 7^5 + 3$
Apollo Okwuone, 12/04
Olathe, KS		      
                    36750 (2.2)
$\dfrac{735}{2\%}$
Dave Jones, 2/07
Coventry, England
                    37650 (2.2)
$\dfrac{753}{2\%}$
Dave Jones, 2/07
Coventry, England
        39064 (3.0)
$\dfrac{5^7 + 3}{2}$
Tyler Krieg, 8/12
Leawood, KS		            
                    50000 (3.0)
$\dfrac{(7+3)^5}{2}$
Allan Hackett, 2/04
Overland Park, KS
          50625 (3.0)
$(75 \times 3)^2$
Zach Burger, 5/05
Overland Park, KS		          
    54432 (3.0)
$(2 \times 3)^5 \times 7$
David Caton, 9/04
Fort Scott, KS		                
        69984 (3.0)
$2^5 \times 3^7$
Bridget Osei-Bonsu, 11/04
Lenexa, KS		            
            78116 (3.0)
$5^7 - 3^2$
Bridget Osei-Bonsu, 12/04
Lenexa, KS		 78117 (3.0)
$5^7 - 2^3$
Phaedra Dowell, 1/05
Roeland Park, KS		   78119 (3.0)
$5^7 - 2 \times 3$
Brooke Jones, 3/04
Olathe, KS		 78120 (3.0)
$5^7 - 3 - 2$
Steve Wilson, 2/07
Raytown, MO
        78124 (3.0)
$5^7 - 3 + 2$
Steve Wilson, 2/07
Raytown, MO		 78125 (3.0)
$5^7 \times (3 - 2)$
Steve Wilson, 2/07
Raytown, MO		 78126 (3.0)
$5^7 + 3 - 2$
Steve Wilson, 2/07
Raytown, MO		       78130 (3.0)
$5^7 + 3 + 2$
Steve Wilson, 2/07
Raytown, MO
  78131 (3.0)
$5^7 + 3 \times 2$
Steve Wilson, 2/07
Raytown, MO		   78133 (3.0)
$5^7 + 2^3$
Phaedra Dowell, 1/05
Roeland Park, KS		 78134 (3.0)
$5^7 + 3^2$
Brooke Jones, 2/04
Olathe, KS		            
          84035 (3.0)
$5 \times 7^{3+2}$
Alyssa Trembly, 4/15
Spring Hill, KS		          
                    85050 (3.2)
$3^5 \times \dfrac{7}{2\%}$
Jordan Pfeiffer, 6/12
Mission Woods, KS
                    85750 (3.0)
$(7 \times 5)^3 \times 2$
David Caton, 10/04
Shawnee, KS
          98415 (3.0)
$27^3 \times 5$
Brittney King, 12/07
Olathe, KS		          
    100842 (3.0)
$7^5 \times 3 \times 2$
Brittney King, 11/07
Olathe, KS		                
                    103200 (3.6)
$\dfrac{7! + 5!}{(3 + 2)\%}$
Long Zhang, 7/12
Lawrence, KS
          109375 (3.0)
$25^3 \times 7$
Kimberly Grosdidier, 12/05
Eudora, KS		          
        113724 (3.0)
$3^7 \times 52$
Brittney King, 11/07
Olathe, KS		            
        117624 (3.2)
$7^{3!} - 5^2$
Phaedra Dowell, 3/05
Roeland Park, KS		            
        117674 (3.2)
$7^{3!} + 5^2$
Phaedra Dowell, 3/05
Roeland Park, KS		            
                    126150 (3.4)
$5^2 \times (7! + 3!)$
Brooke Jones, 5/04
Olathe, KS
                  127449 (3.0)
$357^2$
Tim Murphy, 5/03
Olathe, KS		  
            134456 (3.0)
$7^5 \times 2^3$
Bridget Osei-Bonsu, 11/04
Lenexa, KS		        
                  288369 (3.0)
$537^2$
Brooke Jones, 2/04
Olathe, KS		  
                    343000 (3.0)
$(7 \times 5 \times 2)^3$
Bridget Osei-Bonsu, 12/04
Lenexa, KS
            370386 (3.0)
$57^3 \times 2$
Katie Cooper, 9/04
Stilwell, KS		        
                    393660 (3.2)
$\dfrac{27^3}{5\%}$
Jordan Pfeiffer, 6/12
Mission Woods, KS
      413343 (3.0)
$7 \times 3^{2 \times 5}$
Jordan Pfeiffer, 6/12
Mission Woods, KS		              
                    468750 (3.0)
$2 \times 3 \times 5^7$
Cassandra McLeod, 4/02
Overland Park, KS
        537824 (3.0)
$7^5 \times 32$
Katie Cooper, 8/04
Stilwell, KS		            
                  567009 (3.0)
$753^2$
Brittney King, 9/07
Olathe, KS		  
          588245 (3.0)
$5 \times 7^{3 \times 2}$
Alyssa Trembly, 4/15
Spring Hill, KS		          
    604812 (3.6)
$2 \times 3! + 5! \times 7!$
Brooke Jones, 4/04
Olathe, KS		                
          703125 (3.0)
$5^7 \times 3^2$
Bridget Osei-Bonsu, 12/04
Lenexa, KS		          
                    843750 (3.0)
$75^3 \times 2$
Brittney King, 9/07
Olathe, KS
            984256 (3.0)
$52^3 \times 7$
Brittney King, 12/07
Olathe, KS		        
            1048576 (3.0)
$(7 - 3)^{5 \times 2}$
Allan Hackett, 2/04
Overland Park, KS		        
          1796875 (3.0)
$23 \times 5^7$
Brittney King, 10/07
Olathe, KS		          
                    1866240 (3.0)
$72^3 \times 5$
Brittney King, 9/07
Olathe, KS
          2460375 (3.0)
$(27 \times 5)^3$
Kashmira Sayani, 4/17
Overland Park, KS		          
                    2500000 (3.0)
$5^7 \times 32$
Brittney King, 11/07
Olathe, KS
                    2521050 (3.2)
$7^5 \times \dfrac{3}{2\%}$
Jordan Pfeiffer, 6/12
Mission Woods, KS
        2985984 (3.0)
$(7 + 5)^{3 \times 2}$
Allan Hackett, 2/04
Overland Park, KS		            
  3628821 (3.2)
$(5 \times 2)! + 7 \times 3$
Phaedra Dowell, 1/05
Roeland Park, KS		                  
    8168202 (3.0)
$(7 \times 3)^5 \times 2$
David Caton, 10/04
Shawnee, KS		                
      16974593 (3.0)
$257^3$
Tim Murphy, 5/03
Olathe, KS		              
                    24300000 (3.0)
$(23 + 7)^5$
Katie Cooper, 11/04
Stilwell, KS
  40346721 (3.0)
$27^5 \times 3$
Jordan Pfeiffer, 6/12
Mission Woods, KS		                  
  45054401 (3.0)
$23^5 \times 7$
Phaedra Dowell, 2/05
Roeland Park, KS		                  
          52521875 (3.0)
$(7 \times 5)^{3+2}$
Alyssa Trembly, 4/15
Spring Hill, KS		          
                187149248 (3.0)
$572^3$
Brittney King, 11/07
Olathe, KS		    
        234881024 (3.0)
$32^5 \times 7$
Brittney King, 12/07
Olathe, KS		            
                244140618 (3.2)
$5^{3! \times 2} - 7$
Phaedra Dowell, 5/05
Roeland Park, KS		    
    244140632 (3.2)
$5^{3! \times 2} + 7$
Phaedra Dowell, 5/05
Roeland Park, KS		                
                    341718750 (3.0)
$(5 \times 3)^7 \times 2$
David Caton, 10/04
Shawnee, KS
                425259008 (3.0)
$752^3$
Apollo Okwuone, 12/04
Olathe, KS		    
              1073741817 (3.2)
$2^{3! \times 5} - 7$
Phaedra Dowell, 5/05
Roeland Park, KS		      
  1073741831 (3.2)
$2^{3! \times 5} + 7$
Phaedra Dowell, 5/05
Roeland Park, KS		                  
          1838265625 (3.0)
$(7 \times 5)^{3 \times 2}$
Cassandra McLeod, 5/02
Overland Park, KS		          
              3404825327 (3.2)
$23^7 - 5!$
Phaedra Dowell, 2/05
Roeland Park, KS		      
              3404825567 (3.2)
$23^7 + 5!$
Phaedra Dowell, 2/05
Roeland Park, KS		      
            4146143186 (3.0)
$73^5 \times 2$
Brittney King, 9/07
Olathe, KS		        
                  6103515619 (3.2)
$25^7 - 3!$
Phaedra Dowell, 2/05
Roeland Park, KS		  
  6103515631 (3.2)
$25^7 + 3!$
Phaedra Dowell, 2/05
Roeland Park, KS		                  
            7256313856 (3.6)
$\left( \dfrac{7!}{5!} + 2 \right)^{3!}$
Rabeh Ghadiri, 12/08
Overland Park, KS		        
  34359738361 (3.0)
$2^{35} - 7$
Lucas Rodriguez, 11/04
Lenexa, KS		                  
                    googol (3.4)
$\sqrt[5\% \times .2]{7 + 3}$
Anders Skjäl, 2/04
Turku, Finland
                    googolplex (4.6)
$\antilog\antilog\antilog 2$
$\phantom. \times (5 - 7 + 3)$
Bee Moua, 7/25
Leola, PA

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