PRICING SUPPLEMENT NO. 31                                         Rule 424(b)(3)
DATED:   September 1, 2004                                   File No. 333-109793
(To Prospectus dated November 17, 2003,
and Prospectus Supplement dated November 17, 2003)

                                 $10,668,950,162
                         THE BEAR STEARNS COMPANIES INC.
                           Medium-Term Notes, Series B

Principal Amount: $400,000,000  Floating Rate Notes [x]   Book Entry Notes [x]

Original Issue Date:  9/9/2004  Fixed Rate Notes [ ]      Certificated Notes [ ]

Maturity Date:  9/9/2009        CUSIP#: 073928G56

                                Euroclear and Clearstream Common Code: 020066784

Option to Extend Maturity:      No    [x]
                                Yes   [ ]  Final Maturity Date:


                                              Optional            Optional
                         Redemption           Repayment           Repayment
   Redeemable On          Price(s)             Date(s)            Price(s)
   -------------       --------------       -------------      -----------

        N/A                  N/A                 N/A                 N/A

Applicable Only to Fixed Rate Notes:
- -----------------------------------
Interest Rate:

Interest Payment Dates:

Applicable Only to Floating Rate Notes:
- --------------------------------------

Interest Rate Basis:                    Maximum Interest Rate:  N/A

[ ]  Commercial Paper Rate              Minimum Interest Rate:  N/A

[ ]  Federal Funds Effective Rate

[ ]  Federal Funds Open Rate            Interest Reset Date(s): *

[ ]  Treasury Rate                      Interest Reset Period:  Quarterly

[ ]  LIBOR Reuters                      Interest Payment Date(s): **

[x]  LIBOR Telerate

[ ]  Prime Rate

[ ]  CMT Rate

Initial Interest Rate: 2.13%             Interest Payment Period:  Quarterly

Index Maturity:  Three Months

Spread (plus or minus):  +0.27%

*   On the 9th of each December, March, June and September prior to Maturity.

**  On the 9th of each December, March, June and September, including the
    maturity date.

The distribution of Notes will conform to the requirements set forth in Rule
2720 of the NASD Conduct Rules.